1 Introduction

Water-wave motion is ubiquitous and can be observed most easily on rivers, on canals, on lakes, at the beach and at sea. Understanding water-wave dynamics is important, since waves impacting on shores and structures can have destructive effects, both on the natural barriers protecting human habitation and on man-made infrastructures. Industrial maritime-engineering structures at sea, such as ships, offshore wind-turbines, and offshore oil platforms, need to be sufficiently robust to endure considerable periods under wave impact. Wave impact by extreme or rogue water waves can be particularly detrimental due to their large amplitude and sudden appearance [1], all of which motivates their study. Additionally, such rogue waves are fascinating to watch in their own right, leading to the straightforward question studied here: how can we model and generate maximum-amplitude water waves from incoming waves of equal amplitude on an open or periodic horizontal plane?

A water wave is considered extreme when its height H exceeds twice the significant wave height \(H_s\) of the ambient waves [2]. Wave height therein refers to the vertical distance from the wave trough to the wave crest. The significant wave height \(H_s\) was originally defined as the wave height of the highest third of the ambient waves, a definition later replaced as being four times the standard deviation of the surface elevation [2]. Such rogue waves are statistically rare and concern waves in the tail of the wave-height distribution with wave heights larger than \(2H_s.\) Investigations of extreme or rogue waves involve, on the one hand, observational analysis and simulation of wave statistics [3,4,5]; on the other, extreme-wave heights can be observed and calculated exactly and deterministically when multiple solitonic or cnoidal waves interact as crossing waves, or in seas with the highest wave peaks emerging where the separate linear segments of the wave crests cross in a nonlinear fashion; see e.g. [6].

Exact solutions of such waves have been derived for the Kadomtsev–Petviashvili equation (KPE) [7], which is an asymptotic model equation of the more advanced potential-flow water-wave equations (PFE), with the KPE modelling the free-surface elevation of the water in two horizontal spatial dimensions X and Y and time \(\tau \) (dimensionless variables). Within the KPE, the lateral Y-dependence of the dynamics is asymptotically weak and only unidirectional wave propagation in the X-direction is retained. The PFE are in turn an approximation to the incompressible Navier–Stokes equations with a free water surface, or to the more complex two-phase fluid equations, the latter modelling both air and water as well as their smooth or broken interface(s). Within the present context, the PFE are considered as parent equations representing reality, with possible wave-breaking parameterised in a phenomenological fashion [8]. Various intermediate-complexity asymptotic water-wave models between KPE and PFE exist, concerning either unidirectional or bidirectional wave propagation in the two horizontal dimensions. Bidirectional water-wave models generally require numerical modelling to analyse their solutions. We have in particular considered the bidirectional Benney–Luke equations (BLE) [9, 10], both because the BLE are often used as an interim step to derive and assess solutions of the KPE [11], and because the BLE are readily discretised numerically in a geometric (and, as such, consistent) manner via a straightforwardly available variational principle (VP) [12]. Consistent variational discretisations are deemed important for investigating maximum-wave amplification, because wave amplitude can be preserved in such so-called compatible or geometric numerics, which preservation is tied to energy conservation as well as to other conservation laws. In summary, we remark that the BLE are more advanced and realistic than the KPE, and that the PFE in turn improve upon the BLE.

Inspired by the aforementioned statistical and deterministic extreme-wave analyses, we have investigated how realistic are some extreme-solitonic wave interactions modelled with KPE [13]. For the KPE, we have assessed analytically the maximal attainable amplitude of three interacting solitons, with each individual soliton having an archetypical \(\text {sech}\)-profile amplitude of the same magnitude in its far field. The three-soliton solution is achieved by two pairs of interacting solitons, both of which in separation constitute a travelling-wave O-type (2, 2)-line soliton solution (according to Kodama’s classification [14]) that is steady in its frame of reference. Even though such three-soliton interactions have been hitherto analysed, exact analysis establishing the maximum amplification factor of nine appeared to be missing, so it is presented fully herein. In contrast to the better-known two-soliton interaction with ever-present maximally fourfold amplification, the maximum for the three-soliton case is achieved at one point \((X^*,Y^*,\tau ^*)\) in space and time, since the two pairs of two-soliton solutions collide under an optimal angle and their combination is unsteady. Such an analysis is required, because we seed the BLE with the exact KPE solution at a time prior to the maximum wave-peak occurrence to assess the wave coherence at later times in the more complex BLE. Hence, we are able to assess numerically what maximum amplification is attainable for the more realistic BLE. However, such an approach has at least two shortcomings that the present work now aims to address.

The maximum ninefold amplification is achievable only in the vanishing limit \(\delta \rightarrow 0\) of a small parameter \(\delta \) that is proportional to the difference of the two smallest of three (positive) wavenumber parameters, with the remaining three wavenumbers being equal and opposite to the three main ones. In that limit, the Y-location of the maximum was shown to diverge in our earlier calculations, with \(Y^*\rightarrow -\infty ,\) while \(X^*=\tau ^*=0.\) As a consequence, the numerical simulation of the BLE was seeded with an exact solution of the KPE for a finite-yet-small value of \(\delta \) to avoid this divergence: we took the value \(\delta \approx 0.0014\) in [13]. The KPE seed was taken at a time prior to the growing phase from its intermediate-in-time fourfold to ninefold maximum. That value of \(\delta \) relates to a particular maximum amplification of 8.4 for the KPE, while the numerical simulation of BLE gave an amplification of approximately 7.8.

Consequently, our first goal is to improve the analysis of the maximum amplification and to slow down or to avert divergence of its space-time location \(Y^*.\) Additionally, we aim to show how to seed the BLE and PFE with an initial condition of higher order accuracy in the amplitude and dispersion parameters, respectively, denoted by \(\epsilon \) and \(\mu .\) A key step towards the second goal is to modify the non-periodic velocity potential valid on an infinite plane, such that the dynamics fit within a (more computationally convenient) X-periodic domain, which approximation is possible due the symmetry of the solutions. This minor alteration of the domain in effect turns the solitonic dynamics into cnoidal-wave dynamics in which the maximum amplification in the interior of the domain is hardly (as quantified in the results below) affected by periodicity effects at the boundaries.

As such, our second goal is to numerically assess the maximum amplification attainable in the more realistic PFE and to compare these PFE-based amplifications with new higher order ones using the BLE. Effects that will limit the amplification of the three interacting solitons in the PFE are as follows: dispersive effects will lead to wave-spreading and lower amplitudes; higher amplitude waves will break, thereby lowering wave amplitudes; and the numerical modelling for the PFE will be much more demanding than that for the BLE.

To achieve high-level (as defined below) and consistent simulations of the PFE, we have built a new three-dimensional solver based on implementing a suitable time-discrete variational principle (VP) of potential-flow dynamics directly into the finite-element environment Firedrake for solving partial differential equations [15]. Essentially, a modification of Luke’s variational principle for water waves [16] is formulated. Rather than manually implementing consistent-yet-cumbersome weak formulations of the PFE, Firedrake automatically generates these from the relevant time-discrete VP, whereupon in-built Firedrake solvers and parallelised simulations are readily defined and available for higher order spatial discretisations. Generally, for these demanding nonlinear wave motions, we need at least quadratic-polynomial approximations in space and second-order accuracy in time. The spatial accuracy in Firedrake can be varied easily and independently in the horizontal and vertical directions, and the default-implemented Gauss–Lobatto–Legendre (GLL) polynomials are spectrally accurate. Such a versatile high-level approach—namely Firedrake’s intrinsic use of time-discrete VPs—reduces development time, eradicates manual-derivation errors, and allows straightforward modification to accommodate dynamics within an X-periodic domain. Moreover, the numerical implementation is tightly connected to the mathematical formulation, since the employed unified form language (UFL) is fully decomposable. In plain terms, this means that a nearly phonetic translation of the relevant formulas of the variational principle establishes a working code for the wave dynamics. Both second-order modified-midpoint and Störmer–Verlet time discretisations have been defined and tested for both BLE and PFE. The novel codes were carefully constructed and verified, first for a single travelling soliton, then further developed for the more advanced two- and three-soliton interactions of interest. Within this environment, simulations using over 2 million degrees-of-freedom have allowed the wave science to commence, while solver optimisation remains ongoing. We remark that optimisation relies heavily on PETSc solvers of the relevant linear and nonlinear algebraic systems [17, 18].

In summary, the above-described computer-science approach to solving complicated wave dynamics within Firedrake forms a third novel goal of our work. The VP-based strategy includes and extends earlier work based on discretising the weak formulations following from discretised variational principles, including approaches in which: (i) the mesh is moved according to the free-surface movement [19, 20]; and (ii) the computational domain is fixed after a coordinate transformation and only one element is used in the vertical direction, combined with higher order Lagrange polynomials (implemented within Firedrake) [21, 22]. Both of these previous approaches used first-order piecewise-linear continuous Galerkin polynomials (denoted as CG1, at least in the horizontal), while the current approach runs with the necessary higher order GLL polynomials, typically CG2 or higher.

As a result of the above, the following outlook of the current paper emerges. In Sect. 2, the hierarchy of the three wave models—PFE, BLE, and KPE—is introduced, including the two scalings involved. KPE and BLE are established in different dimensionless coordinates, while we keep the dimensional form of the PFE, essentially using SI-units as (hidden) scaling in the PFE computations. The extended analysis—of the maximum amplification of three-soliton interaction in KPE—is undertaken in Sect. 3, pertaining to our first goal. The numerical modelling tools are set up in Sect. 4 and detail the computer-science novelty, i.e., our third goal. Simulations results of BLE and PFE are discussed, compared, and analysed in Sect. 5, pertaining to our second goal. We finish with concluding remarks in Sect. 6.

2 Hierarchy of Potential-Flow, Benney–Luke, and Kadomtsev–Petviashvili Equations

To create initial conditions for the Benney–Luke equations (BLE) and the potential-flow equations (PFE), we consider the formal link from the PFE via the BLE to the Kadomtsev–Petviashvili equation (KPE). The PFE in dimensional spatial–temporal coordinates \((x,y,z,t\ge t_0)\) and variables are given by

$$\begin{aligned}&\Delta \phi +\partial _{zz}\phi =0\quad \text {in} \ \Omega \subset {\mathbb {R}}^3, \end{aligned}$$
(1a)
$$\begin{aligned}&\partial _t \eta +\nabla \phi \cdot \nabla \eta -\partial _z\phi =0\quad \text {at} \ z=H_0+\eta , \end{aligned}$$
(1b)
$$\begin{aligned}&\partial _t\phi +\frac{1}{2}|\nabla \phi |^2+\frac{1}{2}(\partial _z\phi )^2+g\eta =0\quad \text {at} \ z=H_0+\eta , \end{aligned}$$
(1c)
$$\begin{aligned}&{\textbf{n}}\cdot \nabla \phi =0\quad \text {on} \ \partial \Omega _s,\end{aligned}$$
(1d)
$$\begin{aligned}&\partial _z\phi =0\quad \text {at}\ z=0, \end{aligned}$$
(1e)

where \(\phi =\phi (x,y,z,t)\) is the velocity potential and \(\eta =\eta (x,y,t)\) is the deviation from the rest surface with \(H_0\) the rest depth. Here, the gradient is in two spatial dimensions, \(\nabla =(\partial _x,\partial _y)^T,\) and \(\Delta =\nabla ^2.\) Also, the Laplace equation is satisfied in a three-dimensional domain \(\Omega \) with boundary \(\partial \Omega ,\) with \(\Omega _s\) denoting solid-wall boundaries. We can formally derive the BLE from (1) using the non-dimensional variables

$$\begin{aligned} \widehat{x}&=\frac{\sqrt{\mu }}{H_0}{x},\quad \widehat{y}=\frac{\sqrt{\mu }}{H_0}{y},\quad \widehat{z}=\frac{1}{H_0}{z},\quad \widehat{t}=\frac{\sqrt{gH_0\mu }}{H_0}{t},\nonumber \\ \widehat{\eta }&=\frac{1}{\epsilon H_0}{\eta },\quad \widehat{\phi }=\frac{\sqrt{\mu }}{\epsilon H_0\sqrt{g H_0}}{\phi }, \end{aligned}$$
(2)

with amplitude parameter \(\epsilon =a/H_0\ll 1\) and dispersion parameter \(\mu =(H_0/\lambda )^2\ll 1;\) here, \(\lambda \) is a typical wavelength. Herein, we denote the seabed potential at \(\widehat{z}=0\) by

$$\begin{aligned} \Phi (\widehat{x},\widehat{y},\widehat{t}):=\widehat{\phi }(\widehat{x},\widehat{y},0,\widehat{t}). \end{aligned}$$
(3)

In the derivation of the BLE, e.g., see [12] and references therein, \(\widehat{\phi }\) has a Taylor expansion in terms of \(\widehat{z}\) as follows:

$$\begin{aligned} \widehat{\phi }(\widehat{x},\widehat{y},\widehat{z},\widehat{t})=\Phi (\widehat{x},\widehat{y},\widehat{t})-\frac{\mu }{2}\widehat{z}^2\widehat{\Delta }\Phi +\frac{\mu ^2}{24}\widehat{z}^4\widehat{\Delta }^2\Phi +\mathcal {O}(\mu ^3\widehat{z}^6). \end{aligned}$$
(4)

Upon using (2) and substituting (4) into (1), and after truncating terms up to order \(\mathcal {O}(\epsilon ^2\mu ,\epsilon ^3),\) the BLE emerge—in the non-dimensional coordinates \((\widehat{x},\widehat{y},\widehat{t}\ge \widehat{t}_0)\in {\mathbb {R}}^3\) for some initial time \(\widehat{t}_0\)—as follows:

$$\begin{aligned}&\partial _{\widehat{t}}\widehat{\Phi } - \frac{\mu }{2}\partial _{\widehat{t}}\nabla ^2\widehat{\Phi } + \frac{\epsilon }{2}|\nabla \widehat{\Phi }|^2 + \widehat{\eta } = 0 \quad \text {in} \ \widehat{\Omega }_h \subset {\mathbb {R}}^2, \end{aligned}$$
(5a)
$$\begin{aligned}&\partial _{\widehat{t}}\widehat{\eta } - \frac{\mu }{2}\partial _{\widehat{t}}\nabla ^2\widehat{\eta } + \nabla \cdot \bigl (\left( 1+\epsilon \widehat{\eta }\right) \nabla \widehat{\Phi }\bigr ) - \frac{2}{3}\mu \nabla ^4\widehat{\Phi } = 0 \quad \text {in} \ \widehat{\Omega }_h, \end{aligned}$$
(5b)
$$\begin{aligned}&\mathbf{{n}}\cdot \nabla \widehat{\Phi } = 0 \quad \text {on} \ \partial \widehat{\Omega }_h, \end{aligned}$$
(5c)
$$\begin{aligned}&\mathbf{{n}}\cdot \nabla (\nabla ^2\widehat{\Phi }) = 0 \quad \text {on} \ \partial \widehat{\Omega }_h. \end{aligned}$$
(5d)

System (5) is valid in a two-dimensional domain \(\widehat{\Omega }_h\) with boundary conditions stated at solid-wall boundaries \(\partial \widehat{\Omega }_h\) (hats have been omitted from the gradients). We note that the BLE system has energy defined by

$$\begin{aligned} E(t)=\int _{\widehat{\Omega }_h} \left[ \frac{1}{2}\widehat{\eta }^2 + \frac{1}{2}(1+\epsilon \widehat{\eta }) |\nabla \widehat{\Phi } |^2 + \frac{\mu }{3}(\nabla ^2\widehat{\Phi })^2 \right] {\textrm{d}}\widehat{x}\,{\textrm{d}}\widehat{y}, \end{aligned}$$
(6)

which is conserved over time (for the proof, see [13]). To formally derive the KPE from the BLE, see [6, 12], a new scaling is introduced as

$$\begin{aligned} {X}&= \sqrt{\frac{\epsilon }{\mu }}\left( \frac{3}{\sqrt{2}}\right) ^{1/3}(\widehat{x}-\widehat{t}), \quad {Y} = \frac{\epsilon }{\sqrt{\mu }}\left( \frac{3}{\sqrt{2}}\right) ^{2/3}\,\widehat{y}, \quad {\tau } = \epsilon \sqrt{\frac{2\epsilon }{\mu }}\,\widehat{t}, \end{aligned}$$
(7a)
$$\begin{aligned} \widehat{\Phi }&= \sqrt{\epsilon }\left( \frac{2^5}{3^4}\right) ^{1/6}\left( {\Psi } + {\mathcal {O}}(\epsilon ^2)\right) , \end{aligned}$$
(7b)

with \(\mu ={\mathcal {O}}(\epsilon ^2)\) and additionally

$$\begin{aligned} {u}=\partial _X{\Psi }. \end{aligned}$$
(7c)

A series of transformationsFootnote 1 results in the well-known KPE [7] in “standard” form

$$\begin{aligned} \partial _X\left( 4\partial _{\tau }u + 6u\partial _X u + \partial _{XXX}u \right) + 3\partial _{YY}u = 0. \end{aligned}$$
(8)

According to KPE theory, cf. [14], a (web-soliton) solution can be set up via Hirota’s transformation

$$\begin{aligned} u(X,Y,\tau )\equiv 2 \partial _{XX}\ln \{K(X,Y,\tau )\}, \end{aligned}$$
(9)

in which the function \(K(X,Y,\tau )\) is a Wronskian whose elements satisfy a linear system of equations; this interim function K is defined later for the particular two- and three-soliton interactions investigated analytically and numerically in the ensuing sections.

Given this exact solution of the KPE, we define

$$\begin{aligned} \Psi (X,Y,\tau )\equiv 2 \partial _{X}\ln \{K(X,Y,\tau )\} \end{aligned}$$
(10)

and substitute \(\Psi \) into (7c), whence relations (5a) or (A1) yield the asymptotic BLE solutions

$$\begin{aligned} \widehat{\eta }&\equiv \left( \frac{4}{3}\right) ^{1/3}u +\epsilon \left[ -\left( \frac{2^8}{3^4} \right) ^{1/6}\partial _\tau \Psi -\left( \frac{3^2}{2^4}\right) ^{1/6}\partial _{X}^3\Psi -\left( \frac{2^2}{3^4} \right) ^{1/6}\left( \partial _X\Psi \right) ^2\right] , \end{aligned}$$
(11a)
$$\begin{aligned} \widehat{\Phi }&\equiv \sqrt{\epsilon }\left( \frac{4\sqrt{2}}{9}\right) ^{1/3}{\Psi }, \end{aligned}$$
(11b)

which are accurate to \(\mathcal {O}(\epsilon ^2).\)

Upon employing Eq. (11) for \(\tau =\tau _0\) or \(\widehat{t}=\widehat{t}_0=\tau _0\sqrt{\mu /(2\epsilon )}/\epsilon \) via (7a), the dimensionless initial condition for the BLE emerges from the exact KPE solution, evaluated at a suitable time, as

$$\begin{aligned} \widehat{\eta }_0(\widehat{x},\widehat{y})&= \widehat{\eta }(\widehat{x},\widehat{y},\widehat{t}_0) \quad \text {and}\quad {\Phi }_0(\widehat{x},\widehat{y})=\Phi (\widehat{x},\widehat{y},\widehat{t}_0). \end{aligned}$$
(12)

Moreover, by applying (2) and (4) in (12), the corresponding (asymptotic and dimensional) initial conditions for the PFE at \(t=t_0\) become

$$\begin{aligned} \eta (x,t,t_0)\equiv \eta _0({x},{y})&=\epsilon H_0\widehat{\eta }_0(\widehat{x},\widehat{y}), \end{aligned}$$
(13a)
$$\begin{aligned} \phi _0({x},{y})&= \frac{\epsilon H_0 \sqrt{g H_0}}{\sqrt{\mu }}\left( \widehat{\Phi }_0(\widehat{x},\widehat{y}) { -\frac{\mu }{2}\widehat{z}_0^2\widehat{\Delta }\widehat{\Phi }_0+ \frac{\mu ^2}{24}\widehat{z}_0^4\widehat{\Delta }^2\widehat{\Phi }_0 }\right) , \end{aligned}$$
(13b)

where \(\widehat{z}_0(x,y) = \big (H_0+\eta _0(x,y)\big )/{H_0}\) and \( \phi (x,y,\eta _0(x,y),t_0)\equiv \phi _0(x,y).\) In what follows, we use the initialisations only to \(\mathcal {O}(1)\) in \(\epsilon \) for \(\widehat{\eta },\) and hence to \(\mathcal {O}(\epsilon )\) for \(\eta .\)

3 Maximum Amplification for Three-Soliton Interactions

In this section, we prove the ninefold maximum amplification by extending and correcting the preliminary proof in [13], but without imposing assumptions (as we did in [13]) on the wave shifts. Instead, we choose any shift-inducing constants arising in the solutions in such a way that the time and spatial location at which the maximum occurs remain finite when \(\delta \rightarrow 0.\)

The proof provided unfolds in several stages, as follows:

  • A solution \(u(X,Y,\tau )\) for the KPE will be developed with three equal-amplitude sech-solitons in the far field. This far-field amplitude of each soliton is denoted by \(\tilde{A}.\) We will show that the solution is symmetric around \(Y=Y^*,\) i.e., that its dependence on Y contains only even functions of the argument \(Y-Y^*,\) as can also be discerned from the symmetry of the set-up; in this sense, the present work extends (50, 51) in [13].

  • The candidate point of maximum amplification \((X^*,Y^*,\tau ^*)\) is derived geometrically by considering the five centrelines of the three far-field line-solitons, due to a phase shift in the region of soliton interactions.Footnote 2

  • Given this ansatz for \((X^*,Y^*,\tau ^*),\) and its symmetry around \(Y=Y^*,\) the computer-algebra environment Maple is used to show unambiguously that the maximum amplification factor \(u(X^*,Y^*,\tau ^*)/\tilde{A}\) occurs at this candidate point as a function of a parameter \(\delta \ge 0,\) in the limit \(\delta \rightarrow 0\) of which \(u(X^*,Y^*,\tau ^*)/\tilde{A}\rightarrow 9.\) Additionally, the \(Y^*\)-location remains finite for certain parameters \(a,c\ne 1\) and \(b=1\) involved, in contrast to the \(a=b=c=1\) case considered in [13] which resulted in \(Y^*\rightarrow - \infty \) in the vanishing-\(\delta \) limit.

3.1 Three-Soliton Solution and Its Y-Symmetry

The function \(K(X,Y,\tau )\) used in (9), e.g., see [13] and references therein, reads

$$\begin{aligned} K(X,Y,\tau )&=A_{135}\,e^{\theta _{135}} + A_{246}\,e^{\theta _{246}} + A_{235}\,e^{\theta _{235}} + A_{245}\,e^{\theta _{245}} \nonumber \\&\quad + A_{136}\,e^{\theta _{136}} + A_{146}\,e^{\theta _{146}} + A_{236}\,e^{\theta _{236}} + A_{145}\,e^{\theta _{145}}, \end{aligned}$$
(14)

where \(\theta _{ijk}=(k_i+k_j+k_k)X+(k_i^2+k_j^2+k_k^2)Y-(k_i^3+k_j^3+k_k^3)\tau ,\) in which the eight terms have been ordered pairwise and the six coefficients \(k_i,k_j,k_k\) are defined by (in which we assume three equal-amplitude solitons in the far field)

$$\begin{aligned} k_6&= -k_1 = \frac{1}{2}\tan \theta + \sqrt{\tilde{A}/2} = \sqrt{\tilde{A}}\left( \sqrt{2}+\sqrt{1/2}+\delta \right) , \end{aligned}$$
(15a)
$$\begin{aligned} k_5&= -k_2 = \frac{1}{2}\tan \theta - \sqrt{\tilde{A}/2} = \sqrt{\tilde{A}}\left( \sqrt{1/2} + \delta \right) , \end{aligned}$$
(15b)
$$\begin{aligned} k_4&= -k_3 = \sqrt{\tilde{A}/2}. \end{aligned}$$
(15c)

The coefficients in (14) are functions of the \(k_i\) and are defined as follows:

$$\begin{aligned} A_{135}&= (k_3k_5^2-k_5k_3^2-k_1k_5^2+k_1k_3^2+k_5k_1^2-k_3k_1^2), \end{aligned}$$
(16a)
$$\begin{aligned} A_{235}&= a\,(k_3k_5^2-k_5k_3^2-k_2k_5^2+k_2k_3^2+k_5k_2^2-k_3k_2^2), \end{aligned}$$
(16b)
$$\begin{aligned} A_{136}&= c\,(k_3k_6^2-k_6k_3^2-k_1k_6^2+k_1k_3^2+k_6k_1^2-k_3k_1^2), \end{aligned}$$
(16c)
$$\begin{aligned} A_{236}&= a\,c\,(k_3k_6^2-k_6k_3^2-k_2k_6^2+k_2k_3^2+k_6k_2^2-k_3k_2^2), \end{aligned}$$
(16d)
$$\begin{aligned} A_{145}&= b\,(k_4k_5^2-k_5k_4^2-k_1k_5^2+k_1k_4^2+k_5k_1^2-k_4k_1^2), \end{aligned}$$
(16e)
$$\begin{aligned} A_{245}&= a\,b\,(k_4k_5^2-k_5k_4^2-k_2k_5^2+k_2k_4^2+k_5k_2^2-k_4k_2^2), \end{aligned}$$
(16f)
$$\begin{aligned} A_{146}&= b\,c\,(k_4k_6^2-k_6k_4^2-k_1k_6^2+k_1k_4^2+k_6k_1^2-k_4k_1^2), \end{aligned}$$
(16g)
$$\begin{aligned} A_{246}&= a\,b\,c\,(k_4k_6^2-k_6k_4^2-k_2k_6^2+k_2k_4^2+k_6k_2^2-k_4k_2^2), \end{aligned}$$
(16h)

where a, b, c are positive constants that shift the location of the solution (with \(a=b=c=1\) corresponding to no shift). Using the skew symmetries \(k_1=-k_6,\) \(k_2=-k_5,\) \(k_3=-k_4\) with \(k_6>k_5\ge k_4>0,\) the coefficients (16) can (recalling the pairings in (14)) be simplified as follows:

$$\begin{aligned} A_{135}&= -k_4 k_5^2-k_5 k_4^2+k_6 k_5^2-k_6 k_4^2+k_5 k_6^2+k_4 k_6^2 \end{aligned}$$
(17a)
$$\begin{aligned} A_{246}&= a\,b\,c\, (k_4 k_6^2-k_6 k_4^2+k_5 k_6^2-k_5 k_4^2+k_6 k_5^2-k_4 k_5^2) = a\,b\,c\, A_{135} \end{aligned}$$
(17b)
$$\begin{aligned} A_{235}&= 2\,a\, k_5 (k_5^2-k_4^2) \end{aligned}$$
(17c)
$$\begin{aligned} A_{245}&= 2\,a\,b\, k_5 (k_5^2-k_4^2) = b\, A_{235} \end{aligned}$$
(17d)
$$\begin{aligned} A_{136}&= 2\,c\, k_6 (k_6^2-k_4^2) \end{aligned}$$
(17e)
$$\begin{aligned} A_{146}&= 2\,b\,c\, k_6 (k_6^2-k_4^2) = b\, A_{136} \end{aligned}$$
(17f)
$$\begin{aligned} A_{236}&= a\,c\, (-k_4 k_6^2-k_6 k_4^2+k_5 k_6^2-k_5 k_4^2+k_6 k_5^2+ k_4 k_5^2) \end{aligned}$$
(17g)
$$\begin{aligned} A_{145}&= b\,(k_4 k_5^2-k_5 k_4^2+k_6 k_5^2-k_6 k_4^2 +k_5 k_6^2 -k_4 k_6^2) = b\, A_{236}/(a\,c). \end{aligned}$$
(17h)

The dominant terms in \(K(X,Y,\tau )\) in the far field of the four quadrants are as follows:

$$\begin{aligned}&K_{[1,2]^{-}}\approx A_{135}\,{\textrm{e}}^{\theta _3+\theta _5}\left( {\textrm{e}}^{\theta _1} +\frac{A_{235}}{A_{135}}{\textrm{e}}^{\theta _2}\right) \quad \text {for}\ X<0, Y\rightarrow -\infty \end{aligned}$$
(18a)
$$\begin{aligned}&K_{[5,6]^{-}}\approx A_{135}\,{\textrm{e}}^{\theta _1+\theta _3}\left( {\textrm{e}}^{\theta _5} +\frac{A_{{136}}}{A_{135}}{\textrm{e}}^{\theta _6}\right) \quad \text {for}\ X<0, Y\rightarrow \infty \end{aligned}$$
(18b)
$$\begin{aligned}&K_{[5,6]^{+}}\approx A_{245}\,{\textrm{e}}^{\theta _2+\theta _4}\left( {\textrm{e}}^{\theta _5} +\frac{A_{246}}{A_{245}}{\textrm{e}}^{\theta _6}\right) \quad \text {for}\ X>0, Y\rightarrow -\infty \end{aligned}$$
(18c)
$$\begin{aligned}&K_{[1,2]^{+}}\approx A_{146}\,{\textrm{e}}^{\theta _4+\theta _6}\left( {\textrm{e}}^{\theta _1} +\frac{A_{246}}{A_{146}}{\textrm{e}}^{\theta _2}\right) \quad \text {for}\ X>0, Y\rightarrow \infty . \end{aligned}$$
(18d)

In addition, upon using \(k_3=-k_4,\) the [3, 4] soliton is by design aligned to lie parallel to the Y-axis

$$\begin{aligned} K_{[3,4]} \approx A_{136}\,{\textrm{e}}^{\theta _1+\theta _6}\left( {\textrm{e}}^{\theta _3} + \frac{A_{146}}{A_{136}}{\textrm{e}}^{\theta _4} \right) = A_{136}\,{\textrm{e}}^{\theta _1+\theta _6}\left( {\textrm{e}}^{\theta _3} + {\textrm{e}}^{\theta _4} \right) , \end{aligned}$$
(18e)

in which we have already used that \(b=1,\) such that \(A_{136}=A_{146}.\) Hence, in the far field, the solution \(u(X,Y,\tau )\) is approximated by solitons whose centrelines are determined by the vanishing argument of the soliton solution (see (18) and referring to [13] for further details). Again using the above trio of skew symmetries, these centrelines are given by

$$\begin{aligned}{}[1,2]^-:&\quad -(k_6-k_5)X+(k_6^2-k_5^2)Y+(k_6^3-k_5^3)\tau -\ln {A_{235}/A_{135}} = 0\nonumber \\&\quad \text {for}\ X<0, Y \rightarrow -\infty \end{aligned}$$
(19a)
$$\begin{aligned} ^+:&\quad -(k_6-k_5)X+(k_6^2-k_5^2)Y+(k_6^3-k_5^3)\tau -\ln {A_{246}/A_{146}} = 0 \nonumber \\&\quad \text {for}\ X>0, Y \rightarrow \infty \end{aligned}$$
(19b)
$$\begin{aligned} ^-:&\quad (k_5-k_6)X+(k_5^2-k_6^2)Y-(k_5^3-k_6^3)\tau - \ln {A_{136}/A_{135}}= 0 \nonumber \\&\quad \text {for}\ X<0, Y \rightarrow \infty \end{aligned}$$
(19c)
$$\begin{aligned} ^+:&\quad (k_5-k_6)X+(k_5^2-k_6^2)Y-(k_5^3-k_6^3)\tau - \ln {A_{246}/A_{245}}= 0 \nonumber \\&\quad \text {for}\ X>0, Y \rightarrow -\infty \end{aligned}$$
(19d)
$$\begin{aligned} :&\quad (k_3-k_4)X - (k_3^3-k_4^3)\tau = 0 \Longrightarrow X = k_4^2 \tau . \end{aligned}$$
(19e)

Figure 1 shows a top view of the five centrelines at three different times. The mean of the centrelines \([1,2]^{-}\) and \([5,6]^-,\) as well as the offset \([1,2]^+\) and \([5,6]^+\) centrelines define the same centreline position by adding these line definitions, yielding

$$\begin{aligned} Y^* = {\frac{1}{2}} \frac{\ln {(A_{235}/A_{136})}}{k_6^2-k_5^2}= {\frac{1}{2}} \frac{\ln {(A_{245}/A_{146})}}{k_6^2-k_5^2} = \frac{\ln {\left( \frac{2a k_5(k_5^2-k_{{4}}^2)}{2c k_6(k_6^2-k_4^2)}\right) }}{{2} \left( k_6^2-k_5^2\right) }, \end{aligned}$$
(20)

since \(A_{245}=b A_{235}\) and \(A_{146}=b A_{136}\) from (17), for which we are not per se required to choose \(b=1.\)

Fig. 1
figure 1

Limiting the five centrelines of the three far-field solitons at three times leads to a candidate location of the maximum in space-time \((X,Y,\tau ).\) Blue lines: \(\tau <\tau ^*,\) black lines: \(\tau =\tau ^*,\) red lines \(\tau >\tau ^*.\) The slanted solid/dashed lines are the [1, 2] and [5, 6] solitons, respectively, and the vertical dotted lines are the [3, 4] soliton. The hypothesized maximum is asterisked at time \(\tau ^*\) when and where the black dotted line coincides with the centre of the black diamond. When \(\tau =\tau ^*,\) the edges of the diamond coincide with \(Y^*\) as well as the centre of the diamond and are indicated by the dots; at time \(\tau ^*,\) the centre of the diamond coincides with the X-location of the [3, 4]-dotted line

Full size image

Given the pairings of the \(A_{ijk}\) in (17), the function K in (14) can be rewritten more concisely as

$$\begin{aligned} K(X,Y,\tau )&= A_{135}\,{\textrm{e}}^{(k_4^2+k_5^2+k_6^2)Y}\left( {\textrm{e}}^{(k_5-k_4-k_6)X+(k_4^3+k_6^3-k_5^3)\tau }\right. \nonumber \\&\quad +\left. a\,b\,c\, \, {\textrm{e}}^{(k_4+k_6-k_5)X+(k_5^3-k_4^3-k_6^3)\tau } \right) \nonumber \\&\quad + A_{235}\, {\textrm{e}}^{(2 k_5^2+k_4^2)Y} \left( {\textrm{e}}^{-k_4 X+k_4^3 {\tau }} + b\, {\textrm{e}}^{k_4 X-k_4^3\tau } \right) \nonumber \\&\quad +A_{136}\, {\textrm{e}}^{(2k_6^2+k_4^2)Y}\left( {\textrm{e}}^{-k_4 X +k_4^3\tau }+b \, {\textrm{e}}^{k_4 X -k_4^3\tau } \right) \nonumber \\&\quad +A_{236}\, {\textrm{e}}^{(k_5^2+k_4^2+k_6^2)Y}\biggl ( {\textrm{e}}^{(k_6-k_5-k_4)X+(k_4^3+k_5^3-k_6^3)\tau } \nonumber \\&\quad + \frac{b}{a\,c}\, {\textrm{e}}^{(k_4+k_5-k_6)X+(k_6^3-k_4^3-k_5^3)\tau }\biggr ). \end{aligned}$$
(21)

Using (20) and (17), we find that

$$\begin{aligned} A_{235} = A_{136}\, {\textrm{e}}^{2 (k_6^2-k_5^2)Y^*}, \end{aligned}$$
(22)

using which (21) can be developed further by assigning \(b=1\) and defining \({\textrm{e}}^{-\alpha } \equiv \sqrt{(ac)}.\) We then find

$$\begin{aligned}&K(X,Y,\tau ) = 2 {\textrm{e}}^{(k_4^2+k_5^2+k_6^2)Y}\nonumber \\&\quad \times \Bigl ( A_{135}\,{\textrm{e}}^{-\alpha } \cosh {\left( (k_5-k_4-k_6)X+(k_4^3+k_6^3-k_5^3)\tau +\alpha \right) }\nonumber \\&\quad +2 A_{136}\, {\textrm{e}}^{(k_6^2-k_5^2)Y^*} \cosh {(k_4 X-k_4^3\tau )} \cosh {\left( (k_6^2-k_5^2)(Y-Y^*)\right) }\nonumber \\&\quad + A_{236}\, {\textrm{e}}^{\alpha } \cosh {\left( (k_6-k_5-k_4)X+(k_4^3+k_5^3-k_6^3)\tau -\alpha \right) }\Bigr ), \end{aligned}$$
(23)

cf. the limiting case \(a=b=c=1\) in [13, eqns (50,51)]. Hence, \(K, K_X\) and \(K_{XX}\) are readily found and seen to be proportional to \({\textrm{e}}^{(k_4^2+k_5^2+k_6^2)Y},\) whence (9), when rewritten as

$$\begin{aligned} u(X,Y,\tau ) = 2 (K K_{XX}-K^2_X) /K^2, \end{aligned}$$
(24)

contains only even functions

$$\begin{aligned} 1, \quad \cosh {\left( (k_6^2-k_5^2)(Y-Y^*)\right) } \quad \text {and} \quad \cosh ^2{\left( (k_6^2-k_5^2)(Y-Y^*)\right) } \end{aligned}$$

of \(Y-Y^*,\) so that u is symmetric about \(Y=Y^*,\) on which its maximum must lie.

3.2 Geometric Determination of Candidate Maximum

When the five centrelines given in (19) are plotted in Fig. 1 as functions of time, with the quadrant lines halted at \(Y=Y^*\) for the three cases (of \(\tau ;\) see Fig. 1 caption), the candidate maximum is hypothesized to reside at the intersection of the middle of the black diamond and the black dotted [3, 4] line. Based on this geometric setting, the general expressions for the arguments \((X^*,Y^*,\tau ^*)\) for the case \(b=1\) can be derived as

$$\begin{aligned} Y^*&= \frac{1}{2} \ln {\left( A_{245}/A_{146}\right) }/(k_6^2-k_5^2) = \frac{1}{2} \ln {(A_{235}/A_{136})}/(k_6^2-k_5^2), \end{aligned}$$
(25a)
$$\begin{aligned} \tau ^*&= \frac{1}{2} \ln {\left( \frac{A_{136}A_{246}}{A_{135}A_{146}}\right) } /(k_6^3-k_5^3-k_6 k_4^2+k_5 k_4^2)\nonumber \\&= \frac{1}{2} \ln {\left( ac\right) } /(k_6^3-k_5^3-k_6 k_4^2+k_5 k_4^2), \end{aligned}$$
(25b)
$$\begin{aligned} X^*&= k_4^2\tau ^*. \end{aligned}$$
(25c)

When \(a=b=c=1\) (the case considered in [13]), (17) confirms that the argument \(ac={A_{136}A_{246}}/{A_{135}A_{146}}\) is equal to unity, whence \(\tau ^*=0=X^*.\)

Given this geometric determination of the space-time point of maximum-wave amplification, in what follows the maximum of \(U(X,\tau )\equiv u(X,Y^*,\tau )\) at (25) will be considered. It can be shown both numerically and analytically—solely using \(b=1, X=k_4^2\tau ,\) and \(Y=Y^*\)—that the candidate maximum point is a stationary point with \(U_X=U_\tau =0.\) Details of the partially numerical evaluations will follow after first analysing how \(Y^*\) and \(\tau ^*\) behave when \(\delta \rightarrow 0.\)

To explore the \(Y^*\)-dependency on parameters a, b, c in (20), we consider the following cases:

  1. (i)

    Choose \(a=c=1\) as in [13]: in the limit \(k_5-k_4={\delta }\sqrt{\tilde{A}}\rightarrow 0\) (see (15)), while \(\tau ^*= 0\) remains finite, the value of \(Y_{*}\) diverges as follows:

    $$\begin{aligned} Y^* = \frac{1}{2(k_6^2-k_5^2)} \ln {\left( \frac{\delta \sqrt{\tilde{A}}\, k_5(k_5+k_4)}{k_6(k_6^2-k_4^2)}\right) } \xrightarrow [\delta \rightarrow 0]{} -\infty . \end{aligned}$$
    (26a)
  2. (ii)

    It is possible to choose appropriate values of a and c that remove the divergent singularity in (i) as \(\delta \rightarrow 0\) by noting that \(k_5-k_4 = \delta \sqrt{\tilde{A}}.\) Since

    $$\begin{aligned} Y^*&= \frac{1}{2}\frac{1}{k_6^2-k_5^2} \ln {\Bigl (\frac{a k_5(k_5-k_4)(k_5+k_4)}{c k_6 (k_6^2-k_4^2)}\Bigr )}\nonumber \\&= \frac{1}{2}\frac{1}{k_6^2-k_5^2} \ln {\Bigl (\frac{a \delta \sqrt{\tilde{A}}}{c}\frac{k_5(k_5+k_4)}{k_6 (k_6^2-k_4^2)}}\Bigr ) \nonumber \\ \tau ^*&= \frac{1}{2}\frac{\ln {(ac)}}{(k_6^3-k_5^3-k_6 k_4^2+k_5 k_4^2)}, \end{aligned}$$
    (26b)

    it follows that \(\tau ^*=0\) when \(ac=1\) and \(Y^*=0\) when we furthermore take:

    $$\begin{aligned} a = \sqrt{ \frac{k_6(k_6^2-k_4^2)}{\delta \sqrt{\tilde{A}}\,k_5 (k_5+k_4)} } \quad \text {and}\quad c = \frac{1}{a} = \sqrt{ \frac{\delta \sqrt{\tilde{A}}\,k_5 (k_5+k_4)}{k_6(k_6^2-k_4^2)} }. \end{aligned}$$
    (26c)

    Despite the fact that a diverges when \({\delta \rightarrow 0},\) the relevant coefficients in \(K(X,Y,\tau )\) in (23) remain finite in the limit \(\delta \rightarrow 0,\) as seen from

    $$\begin{aligned} A_{135}&= -k_4 k_5^2-k_5 k_4^2+k_6 k_5^2-k_6 k_4^2+k_5 k_6^2+k_4 k_6^2 \underset{ \delta \rightarrow 0}{\longrightarrow }\ 4 \tilde{A}\sqrt{\tilde{A}/2}, \end{aligned}$$
    (27a)
    $$\begin{aligned} A_{235}&= 2\,a\, k_5 (k_5^2-k_4^2) \underset{\delta \rightarrow 0}{\longrightarrow }\ 0, \end{aligned}$$
    (27b)
    $$\begin{aligned} A_{136}&= 2\,c\, k_6 (k_6^2-k_4^2) \underset{\delta \rightarrow 0}{\longrightarrow }\ 0, \end{aligned}$$
    (27c)
    $$\begin{aligned} A_{236}&= a\,c\, (-k_4 k_6^2-k_6 k_4^2+k_5 k_6^2-k_5 k_4^2+k_6 k_5^2+ k_4 k_5^2) \underset{\delta \rightarrow 0}{\longrightarrow }\ -A_{135}\nonumber \\&=-4 \tilde{A} \sqrt{\tilde{A}/2}. \end{aligned}$$
    (27d)

A plot of the \(Y^*\)-location for the two cases above can be seen in Fig. 2, where the divergence of \(Y^*\) as \(\delta \rightarrow 0\) is clearly demonstrated when \(a=c=1.\)

Fig. 2
figure 2

Location \(Y^*\) of maximum amplification as a function of \(\delta ,\) for cases (i, ii) with \(b=1\) and \(\delta =10^{-7}\)

Full size image

3.3 Computer-Algebra Analysis of Maximum Amplitude

Using (20) to fix \(Y=Y^*\) (which is independent of b), we seek values of \(X^*\) and \(\tau ^*\) that yield stationary points of u in (24). Equivalently, \(u_X(X^*,Y^*,\tau ^*)=0\) and \(u_\tau (X^*,Y^*,\tau ^*)=0\) are to be solved for \(X^*\) and \(\tau ^*.\) Since the denominators of both \(u_X\) and \(u_\tau \) are non-zero, it suffices to determine the roots of their numerators, yielding

$$\begin{aligned} \hat{K}^2 \hat{K}_{XXX} - 3 \hat{K} \hat{K}_X \hat{K}_{XX} + 2\hat{K}_X^3 = 0 \end{aligned}$$
(28a)

and

$$\begin{aligned} \hat{K}^2 \hat{K}_{XX \tau } - \hat{K} \hat{K}_\tau \hat{K}_{XX} - 2\hat{K} \hat{K}_X \hat{K}_{X \tau } + 2\hat{K}_\tau \hat{K}_X^2 = 0, \end{aligned}$$
(28b)

in which \(\hat{K} \equiv K(X,Y^*,\tau ).\) Using the skew symmetry \(k_j = -k_{7-j}\) for \(j=1\ldots 6,\) we have

$$\begin{aligned} \hat{K}&= 2\,\left( b\,{{\textrm{e}}^{k_4 \left( X-k_4^2\tau \right) }}+{{\textrm{e}}^{-k_4 \left( X -k_4^2 \tau \right) }} \right) \left( a\sigma _{{1}}k_5 \left( k_5^2-k_4^2 \right) +c\sigma _{{2}}k_6 \left( k_6^2-k_4^2 \right) \right) \nonumber \\&\quad + \left( k_5+k_6 \right) \sigma _{{3}} \bigg ( \left( k_6+k_4 \right) \left( k_5-k_4\right) \left( b\,{{\textrm{e}}^{\alpha _{{ ppm}} X-\gamma _{{ ppm}}\tau }} +ac\,{{\textrm{e}}^{-\alpha _{{ ppm}}X+\gamma _{{ ppm}}\tau }} \right) \nonumber \\&\quad + \left( k_6-k_4 \right) \left( k_5+k_4 \right) \left( abc\,{{\textrm{e}}^{\alpha _{{ pmp}} X-\gamma _{{ pmp}}\tau }} +{{\textrm{e}}^{-\alpha _{{ pmp}}X+\gamma _{{ pmp}}\tau }} \right) \bigg ) , \end{aligned}$$
(29)

in which

$$\begin{aligned} \alpha _{{ pmp}}&= k_4 -k_5 + k_6, \quad \alpha _{{ ppm}} = k_4 +k_5 - k_6, \end{aligned}$$
(30a)
$$\begin{aligned} \gamma _{{ pmp}}&= k_4^3 -k_5^3 + k_6^3,\quad \gamma _{{ ppm}} = k_4^3 +k_5^3 - k_6^3 \end{aligned}$$
(30b)
$$\begin{aligned} \sigma _1&= {\textrm{e}}^{(k_4^{2}+2\,k_5^{2})Y^*},\quad \sigma _2 = {\textrm{e}}^{(k_4^{2}+2\,k_6^{2})Y^*} \quad \text {and}\quad \sigma _3 = \sqrt{\sigma _1 \sigma _2}. \end{aligned}$$
(30c)

Note that the first bracket in (29) is simply \(2 \cosh \big (k_4(X-k_4^2\tau )\big )\) when \(b=1,\) which is constant when \(X=k_4^2 \tau .\) If, in addition to \(b=1,\) we set \(ac=1,\) the exponential pairs in the second and third lines of (29) also collapse into single \(\cosh \) terms.

Insertion of (29) into (28a) and (28b) is conducted within the Maple computer-algebra environment, whose cumbersome manipulations are not presented here. The substitution \(X=k_4^2 \tau \) in (28a) and (28b) admits progress through substantial algebraic expressions in which \(\tau \) appears only as an exponential argument, whose consistent scaling invites the substitution

$$\begin{aligned} \beta \equiv {\textrm{e}}^{(k_5-k_6)(k_4^2-k_5^2-k_5k_6-k_6^2)\tau }, \end{aligned}$$
(31)

using which both (28a) and (28b) become sixth-order polynomials in \(\beta .\) Unfortunately, these sextics do not appear to be solvable for \(\beta \) for general a, b, c,  but, when \(b=1,\) it can be shown using Maple that \(U_X\) in (28a) factorises into the product of quadratic and quartic polynomials in \(\beta ,\) the former yielding \(\beta = \pm \sqrt{ac}.\) Hence, using the positive root consistent with (31), when \(b=1,\) an explicit solution of (28a) is

$$\begin{aligned} \tau ^* = \frac{\ln (ac)}{2(k_5-k_6)(k_4^2-k_5^2-k_5k_6-k_6^2)} \quad \text {and}\quad X^* = k_4^2\tau ^*, \end{aligned}$$
(32)

and it transpires that this pair also solves (28b). If additionally \(ac=1,\) it follows that \(X^*=\tau ^*=0\) (cf. “case (ii)” above). That is, (20) and (32) together give one turning point of u,  i.e., \(\nabla u(X^*,Y^*,\tau ^*)=0,\) though without being able to find the roots \(\beta \) of the above-mentioned quartic polynomial, we may be missing other such points, even when \(b \ne 1.\) We note that (32) is equivalent to the pair (25b) and (25c) when \(b=1,\) thereby corroborating the above geometric argument. It is possible to make progress asymptotically when \(b \approx 1,\) but in this case even the leading-order perturbation to ac in the numerator of (32) is an unenlightening cumbersome combination of all parameters so far defined.

To confirm that \(u(X^*,Y^*,\tau ^*)\) is a maximum, we additionally invoke the principal-minors theorem (PMT) [24, 25] to show both that \(u_{XX}<0\) and that the determinant of the Hessian satisfies

$$\begin{aligned} H(u) \equiv u_{XX}u_{\tau \tau }-u_{X\tau }u_{\tau X} = u_{XX}u_{\tau \tau }-u_{X\tau }^2 >0, \end{aligned}$$
(33)

in which

$$\begin{aligned} u_{XX}&= \big ( 2\,\hat{K}_{{{XXXX}}} \hat{K}^{3} - ( 8\,\hat{K}_{{X}}\hat{K}_{{{XXX}}}+6\,{\hat{K}_{{{XX}}}}^{2}) \hat{K}^{2} \big .\nonumber \\&\quad \big . + 24\,\hat{K} \hat{K}_{{{XX}}}{\hat{K}_{{X}}}^{2}-12\,{\hat{K}_{{X}}}^{4} \big ) /\hat{K}^4, \end{aligned}$$
(34a)
$$\begin{aligned} u_{X\tau }&=\big ( 2\,\hat{K}_{{{XXX\tau }}} \hat{K}^{3} - ( 2\,\hat{K}_{{\tau }}\hat{K}_{{{XXX}}}+6\,\hat{K}_{{X}}\hat{K}_{{{XX\tau }}}+6\,\hat{K}_{{{X\tau }}}\hat{K}_{{{XX}}} ) \hat{K}^{2} \big .\nonumber \\&\quad \big . + ( 12\,\hat{K}_{{\tau }}\hat{K}_{{X}}\hat{K}_{{{XX}}}+12\,\hat{K}_X^2 \hat{K} _{{{X\tau }}} ) \hat{K} -12\,\hat{K}_{{\tau }}\hat{K}_X^3\big )/ \hat{K}^4, \end{aligned}$$
(34b)
$$\begin{aligned} u_{\tau \tau }&=\big (2\,\hat{K}_{{{XX\tau \tau }}}\hat{K}^{3} - (4\,\hat{K}_{{\tau }}\hat{K}_{{{XX\tau }}}+2\,\hat{K}_{{{\tau \tau }}} \hat{K}_{{{XX}}}+4\,\hat{K}_{{X}}\hat{K}_{{{X\tau \tau }}} + 4\,\hat{K}_{X\tau }^2 ) \hat{K}^2 \big .\nonumber \\&\quad \big . + ( 4\,\hat{K}_\tau ^2 \hat{K}_{{{XX}}}+16\,\hat{K}_{{\tau }}\hat{K}_{{X}} \hat{K}_{{{X\tau }}}+4\,\hat{K}_{{{\tau \tau }}}\hat{K}_X^2 ) \hat{K}-12\,\hat{K}_\tau ^{2}\hat{K}_X^2 \big )/\hat{K}^4. \end{aligned}$$
(34c)

Asymptotic approximations of \(u_{XX}\) and H(u) are non-trivial to determine due to the appearance of \(\delta \) in both rational functions of surds and rational-function powers thereof. However, via judicious iterative substitutions in Maple, one can establish the following limiting forms:

$$\begin{aligned} \frac{u_{XX}}{\tilde{A}^2 }&= -81+176\,\sqrt{3}\root 4 \of {2}\sqrt{\delta }-436\,\sqrt{2}\delta + {\mathcal {O}}(\delta ^{3/2}), \quad \delta \rightarrow 0 \end{aligned}$$
(35a)
$$\begin{aligned} \frac{H(u)}{\tilde{A}^6 }&= 5184\,(\sqrt{3}\root 4 \of {2}\sqrt{\delta }-6\,\sqrt{2}\delta ) + {\mathcal {O}}(\delta ^{3/2}), \quad \delta \rightarrow 0 \end{aligned}$$
(35b)
$$\begin{aligned} \frac{u_{XX}}{\tilde{A}^2 }&= -3-\frac{9}{{\delta }^{2}}+{\frac{1953}{32\,{\delta }^{4}}}-{\frac{639\, \sqrt{2}}{4\,{\delta }^{5}}} + {\mathcal {O}}(\delta ^{-6}), \quad \delta \rightarrow \infty \end{aligned}$$
(35c)
$$\begin{aligned} \frac{H(u)}{\tilde{A}^6 }&= 18\,{\delta }^{4}+72\,\sqrt{2}{\delta }^{3}+81\,{\delta }^{2}-324\, \sqrt{2}\delta + {\mathcal {O}}(1), \quad \delta \rightarrow \infty . \end{aligned}$$
(35d)

Expressions (35) are compared in Fig. 3 against numerically computed values of \(u_{XX}\) and H(u). The combination of these expressions and the numerics confirms that \(-81\tilde{A}^2< u_{XX}<-3\tilde{A}^2\) and \(H(u)>0\) for all \(\delta , \tilde{A} >0,\) thereby satisfying the conditions of the PMT and proving that \(u(X^*,Y^*,\tau ^*)\) is indeed a maximum; this finding is qualitatively independent of a and c. It remains only to show that it is the global maximum.

Fig. 3
figure 3

Log–log plot showing comparison of numerically computed \(-u_{XX}/\tilde{A}^2\) (red line) and \(H(u)/\tilde{A}^6\) (blue line) with (higher order forms of) asymptotic approximations (35a) (red discs), (35b) (blue filled squares), (35c) (red circles), and (35d) (blue open squares). See text for discussion

Full size image

Using the starred coordinates, the maximum-amplification factor \(u(X^*,Y^*,\tau ^*)/\tilde{A}\) can be approximated asymptotically as \(\delta \rightarrow 0.\) When \(ac=1,\) equivalently \(\tau ^*=0,\) the asymptotics are straightforward though mechanical. However, when \(ac \ne 1,\) (15) and (32) introduce exponentials of rational functions of \(\delta ,\) of which asymptotic approximation is far more subtle, but still negotiable. It transpires that the amplification factor is dependent upon only \(\delta ,\) the exact expression being

$$\begin{aligned} \frac{u(X^*,Y^*,\tau ^*)}{\tilde{A}} = \frac{20 \alpha _1 \alpha _2 + \alpha _3}{(\alpha _1 + \alpha _4)^2} , \end{aligned}$$
(36)

in which

$$\begin{aligned} \alpha _{{1}}&= \sqrt{\delta \,( 3\,\sqrt{2}+2\,\delta ) ( 3\,\sqrt{2}\delta +{\delta }^{2}+4 ) ( \sqrt{2}+2\,\delta ) ( \sqrt{2}+\delta )}, \end{aligned}$$
(37a)
$$\begin{aligned} \alpha _{{2}}&=\sqrt{2}+{\frac{9\,\sqrt{2}{\delta }^{2}}{5}}+{\frac{ 17\,\delta }{5}}+{\frac{3\,{\delta }^{3}}{5}}, \end{aligned}$$
(37b)
$$\begin{aligned} \alpha _{{3}}&= \left( 144\,{\delta }^{5}+824\,{\delta }^{3}+320\,\delta \right) \sqrt{2}+24\,{\delta }^{6}+686\,{\delta }^{4}+1040\,{\delta }^{2}+72, \end{aligned}$$
(37c)
$$\begin{aligned} \alpha _{{4}}&=6\,\sqrt{2}{\delta }^{2}+2\,{\delta }^{3}+2\,\sqrt{2}+10\,\delta , \end{aligned}$$
(37d)

and of which the asymptotic approximation as \(\delta \rightarrow 0\) is

$$\begin{aligned} 9-8\,\sqrt{3}\root 4 \of {2}\sqrt{\delta }+16\,\sqrt{2}\delta -{\frac{ 19\,{2}^{3/4}\sqrt{3}}{3}{\delta }^{{3/2}}}+8\,{\delta }^{2}-{ \frac{23\,\sqrt{3}\root 4 \of {2}}{72}{\delta }^{{5/2}}} + {\mathcal {O}}(\delta ^3). \end{aligned}$$
(38)

Figure 4 compares the exact and approximate amplification factors, between which the difference is indiscernible on this scale as \(\delta \rightarrow 0,\) at which limit a non-logarithmic replot (not presented) of Fig. 4 shows cusped behaviour, completing the proof that \(u(X^*,Y^*,\tau ^*)/\tilde{A}\) is the unique maximum, and that it is independent of a,  c,  and \(\tilde{A}.\) Independent computations conducted in Python fully corroborate numerically all theoretical results derived in this section.

Fig. 4
figure 4

Semilog plot of amplification factor, as a function of \(\delta ,\) given by exact formula (36) (red line) and its asymptotic approximation (38) (blue discs). These curves all have \(b=1\) and are independent of a,  c and \(\tilde{A}.\) Larger values of \(\delta \) (e.g., \(\delta >0.1\)) are accommodated to illustrate the range of validity of the asymptotic result

Full size image

4 Numerical Modelling

Both the Benney–Luke equations (BLE) and potential-flow equations (PFE) are approximated in space and time by discretisation of their respective variational principles (VPs). We need a numerical approach with a suitable computational domain given that the interactions of two and three solitons of equal far-field amplitudes are defined on an infinite horizontal plane. We denote these two- and three-solution interactions and simulations by SP2 and SP3 respectively, with details of two-soliton solutions SP2 defined in [13]. One approximation is to consider a domain periodic in the x-direction and, given the y-symmetry of the solutions, a channel with walls in the y-direction instead of an infinite domain. However, when we plot the two-soliton and three-soliton solutions for \(\eta ,\) \(\Phi \) or \(\phi \) and the three velocity components \((u_1,u_2,u_3),\) we note that \(\phi \) and \(\Phi \) are not periodic in x;  see Figs. 14 and 5. To resolve this issue, we partition the velocity potential in each of cases SP2 and SP3 into a background part and an x-periodic perturbation. For the PFE, the partition is

$$\begin{aligned} \phi (x,y,z,t) = U_0(y,z)x+\tilde{\phi }(x,y,z,t), \end{aligned}$$
(39)

with a (y, z)-dependent static function \(U_0(y,z),\) and, similarly, for the BLE the partition is

$$\begin{aligned} \Phi (x,y,t) = U_0(y)x+\tilde{\Phi }(x,y,t), \end{aligned}$$
(40)

wherein the variables are scaled according to (2), but, for simplicity, the hats are generally (unless confusion arises) dropped in (40) and hereafter.

The original and partitioned VPs for the PFE are respectively

$$\begin{aligned} 0&= \delta \int _{0}^T \iint _{\Omega _h}\int _0^{h(x,y,t)} \partial _t\phi + \frac{1}{2}|\varvec{\nabla }\phi |^2 + g(z-H_0) \,{\mathrm d}z\,{\mathrm d}x\,{\mathrm d}y\,{\mathrm d}t \end{aligned}$$
(41a)
$$\begin{aligned} 0&= \delta \int _{0}^T \iint _{\Omega _h}\int _0^{h(x,y,t)} \partial _t\tilde{\phi }+ \frac{1}{2}(U_0+\partial _x\tilde{\phi })^2 +\frac{1}{2}\left( x{\partial _y U_0}+ \partial _y\tilde{\phi }\right) ^2\nonumber \\&\quad +\frac{1}{2}\left( x{\partial _z U_0}+\partial _z\tilde{\phi }\right) ^2 + g(z-H_0) \,{\mathrm d}z\,{\mathrm d}x\,{\mathrm d}y\,{\mathrm d}t, \end{aligned}$$
(41b)

both of which are defined on \(\Omega _h=\{(x,y)\in [x_1,x_2]\times [y_1,y_2] \}\) for appropriately chosen values of \(x_1,\) \(x_2,\) \(y_1\) and \(y_2\) that render the perturbed velocity potential x-periodic. The original and partitioned VPs for the BLE are respectively

$$\begin{aligned} 0&=\delta \int _0^T\iint _{\Omega _h} \eta \partial _t\Phi - \Phi \partial _t\eta +\frac{\mu }{2} \nabla \eta \cdot \nabla \partial _t\Phi -\frac{\mu }{2} \nabla \Phi \cdot \nabla \partial _t\eta \nonumber \\&\quad +\frac{1}{2} (1+\epsilon \eta )|\nabla \Phi |^2 +\frac{1}{2}\eta ^2 + \mu \Bigl (\nabla q\cdot \nabla \Phi -\frac{3}{4} q^2\Bigr ) \,{\mathrm d}x\,{\mathrm d}y\,{\mathrm d}t\end{aligned}$$
(42a)
$$\begin{aligned} 0&=\delta \int _0^T\iint _{\Omega _h} \eta \partial _t\tilde{\Phi }- \tilde{\Phi } \partial _t\eta +\frac{\mu }{2} \nabla \eta \cdot \nabla \partial _t\tilde{\Phi } -\frac{\mu }{2} \nabla \tilde{\Phi }\cdot \nabla \partial _t\eta \nonumber \\&\quad +\frac{1}{2} (1+\epsilon \eta )|\nabla (U_0 x+\tilde{\Phi })|^2 +\frac{1}{2} \eta ^2\nonumber \\&\quad + \mu \Bigl (\nabla q\cdot \nabla (U_0 x+ \tilde{\Phi })-\frac{3}{4} q^2\Bigr ) \,{\mathrm d}x\,{\mathrm d}y\,{\mathrm d}t. \end{aligned}$$
(42b)

Using the above VP (42b), or the partitioning (40) with the BLE (5), the reformulated equations for \(\{\eta ,\tilde{\Phi }\}\) become

$$\begin{aligned}&\partial _t\tilde{\Phi } - \frac{\mu }{2}\partial _t\nabla ^2\tilde{\Phi } +\frac{\epsilon }{2}\left|\nabla (U_0x+\tilde{\Phi })\right|^2 + \eta = 0, \end{aligned}$$
(43a)
$$\begin{aligned}&\partial _t\eta - \frac{\mu }{2}\partial _t\nabla ^2\eta + \nabla \cdot \left\{ \left( 1+\epsilon \eta \right) \nabla (U_0x+\tilde{\Phi })\right\} \nonumber \\&\quad - \frac{2\mu }{3}\nabla ^4(U_0x+\tilde{\Phi }) = 0. \end{aligned}$$
(43b)

The numerical models comprise discretisations of time-discrete versions, of the two partitioned VPs above, within the finite-element environment Firedrake [15], in which the VP for the PFE is first transformed to a fixed domain. The spatial discretisation for the PFE can be varied separately in the horizontal and (transformed) vertical directions using a vertically extruded mesh by either varying the (quadrilateral or line) elements and/or varying the Gauss–Lobatto–Legendre (GLL) polynomial order. We denote the polynomials of order \(n_h\) (horizontal) and \(n_v\) (vertical) by \({\textrm{CG}}n_h\) or \({\textrm{CG}}n_v,\) respectively, the notation CG reflecting the Continuous Galerkin approach: the lowest orders used are \({\textrm{CG}}2\) in both horizontal and vertical (with \(n_h=n_v=2\)). The respective weak formulations are subsequently generated automatically within Firedrake using the “functional” command derivative (see 17.5.1 in [26] and 6.4 in [27]). This approach yields the functional derivatives analytically—without using the divergence theorem—such that the first-order derivatives in the VPs are preserved, acting on the velocity potential and/or its test function, the latter equivalent to the variational derivatives \(\delta \Phi ,\delta \eta ,\delta q\) or \(\delta \phi ,\delta h.\) This entirely variational-based procedure reduces both development time and the introduction of human error given that, for the PFE, the weak formulations are algebraically much more cumbersome than for the basic VP as a result of both explicit mesh motion and implicit mesh “motion” induced by the coordinate transformations that render static the fixed computational domain. The time-discrete VPs and further details on the coordinate transformation are relegated to Appendix C.

4.1 Periodisation of Soliton Dynamics

We have shown that the BLE (5) and PFE (1) require modificationFootnote 3 to accommodate simulation in the proposed x-periodic domain. The potential-flow velocity potential is therefore partitioned into a static, non-periodic part and a dynamic, periodic part. Consequently, the velocity-potential perturbations \(\tilde{\phi }\) (in the PFE) and \(\tilde{\Phi }\) (in the BLE) are by construction made periodic in the x-direction. We now describe how we impose initial conditions for \(\eta _0,\) \(\tilde{\Phi }_0\) and \(\tilde{\phi }_0.\)

First, for the BLE, a function \(\eta _0\) is defined by \(\widehat{\eta }\) in (11a) evaluated at \(\epsilon =0\) as

$$\begin{aligned} \eta (x,y,t_0) \equiv \eta _0(x,y) = 2\Bigl (\frac{4}{3}\Bigr )^{1/3}\partial _{XX}\ln \bigl (K(X,Y,\tau _0)\bigr ). \end{aligned}$$
(44)

The moving three-line-soliton (SP3) solution and the interacting two-line soliton solution (SP2) investigated on an infinite plane, respectively, have an X-shaped and Y-shaped structure, which allows definition of a (nearly) periodic analogue for numerical simulation in an x-periodic domain. Snapshots of the SP3 solution, augmented with coordinate pairs \((x_1,y_1),\) \((x_2,y_1),\) \((x_2,y_2)\) and \((x_2,y_2)\) outlining the prospective periodic domains, are shown in Fig. 5.

Fig. 5
figure 5

Top view of a sample initial condition and resulting velocity components for three-soliton interactions (SP3), calculated via (44) and (50) and shown at \(\hat{t}=-200\) for \(\epsilon =0.05,\) \(\mu =\epsilon ^2,\) \(H_0=20~\text {m}\)

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Because, as \(x\rightarrow \pm \infty ,\) the function \({\eta }_0\) returns to zero (as with \({\textrm{sech}}\)-shaped soliton profiles), we can choose \(x_1,\) \(x_2,\) and \(y_2,\) such that \({\eta }_0(x_1,y_2)={\eta }_0(x_2,y_2)\) and \({\eta }_{0x}(x_1,y_2)=0={\eta }_{0x}(x_2,y_2)\) as in [13]. Using the \({\eta }_0\) defined on an infinite plane, the periodic variation \(\eta _0\) thereof is defined as

$$\begin{aligned} \eta _0\big ( x+n(x_2-x_1),y \big ) = {\eta }_0(x,y), \end{aligned}$$
(45)

where \(x\in [x_1,x_2)\) and \(n\in {\mathbb {Z}}.\) Combining (10) and the velocity potential (11b), the velocity potential reads

$$\begin{aligned} \Phi _0(x,y,t_0) \equiv {\Phi }_0(x,y) = 2\sqrt{\epsilon }\, \Bigl (\frac{4\sqrt{2}}{9}\Bigr )^{1/3}\partial _{X}\ln \bigl (K(X,Y,\tau _0)\bigr ). \end{aligned}$$
(46)

Hence, we find that \({\Phi }_{0x}(x_1,y)={\eta }_{0}(x_1,y)={\eta }_{0}(x_2,y)={\Phi }_{0x}(x_2,y),\) such that \({\Phi }_0\) has a \(\tanh \)-shaped profile in the x-direction. Consequently, \({\Phi }_0(x_1,y)\) and \({\Phi }_0(x_2,y)\) cannot be equal. We, therefore, require the function \(U_0(y)\) in (40) to be such that \(\tilde{\Phi }_0(x_1,y)=\tilde{\Phi }_0(x_2,y)\) becomes periodic or equivalently \({\Phi }_0(x_1,y)-U_0(y)x_1={\Phi }_0(x_2,y)-U_0(y)x_2.\) Given this function and the previous two equations, we derive that

$$\begin{aligned} U_0(y) = \frac{{\Phi }_0(x_2,y)-{\Phi }_0(x_1,y)}{x_2-x_1}. \end{aligned}$$
(47a)

Consequently, the periodic initial condition for \(\tilde{\Phi }\) may be defined by

$$\begin{aligned} \tilde{\Phi }_0\big ({x}+n(x_2-x_1),y\big ) = {\Phi }_0(x,y)-U_0(y){x}, \end{aligned}$$
(48)

where \({x}\in [x_1,x_2)\) and \(n\in {\mathbb {Z}}.\) Recall that the hats for coordinates in the BLE domain have been dropped, such that \(x_{1,2}, y_{1,2}\) in BLE and PFE differ by scaling factors.

The procedure for the PFE is similar when it is undertaken with the \(\phi (x,y,z,t_0)\equiv \phi _0(x,y,z)\) expression. To wit, the above procedure for BLE is thus replaced by

$$\begin{aligned} U_0(y,z) = \frac{\phi _0(x_2,y,z)-\phi _0(x_1,y,z)}{x_2-x_1}, \end{aligned}$$
(49a)

while using (11) up to \(\mathcal {O}(\mu ^2)\) or \(\mathcal {O}(\mu ^4).\) Consequently, the periodic initial condition for the velocity potential is defined by

$$\begin{aligned} \tilde{\phi }_0(x+n(x_2-x_1),y,z)={\phi }_0({x},y,z)- U_0(y,z){x}, \end{aligned}$$
(50)

where \(x\in [x_1,x_2)\) and \(n\in {\mathbb {Z}}.\) The z-dependence of \(\tilde{\phi }\) follows automatically by solving the (discretised) Laplace equation but in the (transformed) equations or variational principle the substitution \(\phi (x,y,z,t)=U_0(y,z)x+\tilde{\phi }(x,y,z,t)\) is made. A difference with the approach for the BLE is that \({\Phi }\) is defined at the bottom \(z=0,\) while for the PFE, the prognostic

$$\begin{aligned} {\psi }(x,y,t)&\equiv \phi (x,y,h(x,y,t),t)\nonumber \\&= U_0(y,h(x,y,t)) x+\tilde{\phi }(x,y,h(x,y,t),t)\nonumber \\&= U_0(y,h(x,y,t)) x+ \tilde{\psi }(x,y,t) \end{aligned}$$
(51)

is defined at the free surface \(z=h(x,y,t).\) Note that the functions \(U_0(y)\) and \(U_0(y,z)\) differ between BLE and PFE due to both the different scalings as well as the inclusion of higher order \(\mathcal {O}(\mu ),\mathcal {O}(\mu ^2)\) terms in \(\Phi _0(x,y)\) and \(\phi _0(x,y,z).\) Further details on the particular expressions for interacting solitons are provided in Appendix B.

5 Maximum Amplification of Three Interacting Solitons with Equal Far-Field Amplitudes

A series of numerical simulations of the three-soliton interactions have been undertaken with both the variational Benney–Luke and potential-flow discretisations. Each simulation uses a different nonlinearity or amplitude measure, \(\epsilon =0.01\) and/or 0.05,  based on the scaling of the Benney–Luke equations (BLE). As initial conditions, we seed both sets of equations with the exact solutions of the Kadomtsev–Petviashvili equation (KPE) we have derived for three interacting line-solitons (denoted by KPE-SP3) at a time before the growth to the maximum occurs in space and time. The goal of the simulations is to assess what the maximum amplification can become in these more accurate or faithful Benney–Luke and potential-flow models of water-wave dynamics. For the exact KPE-SP3 solution used in this initialisation, we take \(\delta = 10^{-5}, 10^{-10},\) for which the computations underlying Fig. 4 reveal an amplification factor of up to 8.9998,  whose error relative to the ninefold theoretical maximum is \(1.8 \times 10^{-3}\%.\) First, simulations for the BLE are presented and, subsequently, simulations for the potential-flow equations (PFE); finally, a brief comparison between these simulations is undertaken. We emphasize that the use of variational space-time discretisations means that there is no loss in wave amplitude due to numerical artefacts.

5.1 Amplification in the Benney–Luke Equations

Table 1 Solution and numerical parameter values employed in the simulations of the BLE for the three-soliton interactions (BLE-SP3)
Full size table

Table 1 shows the parameter values used for the BLE simulations; to highlight a few of them, we use a non-dimensional KPE-amplitude of \(\tilde{A}\approx 0.4543,\) yielding a BLE amplitude of \(\widehat{A}=0.5,\) for each of the nonlinearity parameters \(\epsilon =0.01\) and 0.05. The respective simulations run in the time intervals \([-110,30]\) (for \(\epsilon =0.01\)) and in \([-60,20]\) (for \(\epsilon =0.05\)), and have been undertaken on Leeds arc4-HPC on (typically) 40 cores, requiring, respectively, 919 minutes and 1764 minutes to run. The time step was chosen by trial and error to obtain a stable simulation, as judged by monitoring the energy evolution over time. GLL basis functions CG4 were used, yielding formal fifth-order (spectral) spatial accuracy. A second-order Störmer–Verlet (SV) time-stepping scheme was chosen.

Fig. 6
figure 6

Initial conditions for BLE-SP3 simulation with \(\epsilon =0.01,0.05.\) (a, e) \(\widehat{\eta }_0\) and (b, f) total \(\widehat{\Phi }_0,\) given at times \(\widehat{t}_0=-110\) (for \(\epsilon =0.01\)) and \(\widehat{t}_0=-60\) (for \(\epsilon =0.05\)), respectively. Panels (c, d) and (g, h) show cross-sections of \(\widehat{\eta }_0\) at the far-field \(\widehat{y}=24,\) and on the lower boundary \(\widehat{y}=\widehat{y}_*=0,\) respectively

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The initial conditions for \(\eta \) and \(\Phi \) for the BLE simulations with \(\epsilon =0.01\) (at time \(t=-110\)) and \(\epsilon =0.05\) (at time \(t=-60\)) are shown in Fig. 6. The initial times were chosen, such that they are prior to the time when the maximum in the exact KPE solution starts to grow from \(4\tilde{A},\) caused by the pair of two-soliton interactions to its KPE maximum of nearly \(9\tilde{A}.\)

Fig. 7
figure 7

Numerical three-soliton solution for the Benney–Luke equations (BLE-SP3) for \(\epsilon =0.01\) at times \(\widehat{t} =-30\) (left-hand side panels) and \(\widehat{t} = 0\) (right-hand side panels), respectively. Panels a and b show the surface elevation \(\widehat{\eta }\) which is seen to attain a global maximum at \(\widehat{t}=-30\). Panels c and d show that the height of \(\widehat{\eta }\) in the far-field \(\widehat{y} = 24\) is close to the initial height 0.5 at both times. Panels e and f show the potential \(\tilde{\Phi }\)

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Fig. 8
figure 8

Results obtained for a three-soliton simulation of the Benney–Luke equations (BLE-SP3) with \(\epsilon =0.01,\) employing finite-element polynomials of order CG4. The red dashed line is computed from the rescaled KPE solution as a reference. Panels show: a maximum values of \(\widehat{\eta }\) over time; b \(\widehat{A}:=\max _{\widehat{x}}(\widehat{\eta }(\cdot ,\widehat{y}=24,\widehat{t}));\) c amplification \(\max (\widehat{\eta })/\widehat{A}\) against time; and d energy deviation \(E(\widehat{t})-E(\widehat{t}_0)\) in time, where \(E(\widehat{t})\) is defined in (6) for the BLE system and \(\widehat{t}_0=-110.\) The energy deviation from \(E(\widehat{t}_0) = 9.227\) is of the order of \(10^{-4},\) suggesting that the numerics are stable

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Fig. 9
figure 9

Results obtained for three-soliton simulations of the Benney–Luke equations (BLE-SP3) with \(\epsilon =0.05,\) employing finite-element polynomials of order CG4. The red dashed line is computed from the rescaled KPE solution as a reference. Panels show: a maximum values of \(\widehat{\eta }\) over time; b \(\widehat{A}:=\max _{\widehat{x}}(\widehat{\eta }(\cdot ,\widehat{y}=24,\widehat{t}));\) c amplification \(\max (\widehat{\eta })/\widehat{A}\) against time; and d energy deviation \(E(\widehat{t})-E(\widehat{t}_0)\) in time for simulation BLE-SP3, where \(E(\widehat{t})\) is defined in (6) for the BLE system and \(\widehat{t}_0=-60.\) The energy deviation from \(E(\widehat{t}_0) = 24.251\) is of the order of \(10^{-4}\) except around \(\widehat{t}_0,\) which implies that the numerical scheme is stable

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A pair of solution snapshots for the BLE simulation with \(\epsilon =0.01\) is shown in Fig. 7, together with cross-sections at \(\hat{y}=24\) at BLE times \(\{-30,0\}.\) These results give impressions of the overall solution fields including the minor oscillations of small-scale travelling waves shed off the main three solitons. Further results for this BLE simulation in Fig. 8a show the maximum-wave amplitude in the domain in time. It has a maximum of nearly 4.5 around BLE time \(-50,\) whereafter the maximum slightly decreases and swings up. Figure 8b shows the far-field amplitude \(\widehat{A}=\max _{\hat{x}}\widehat{\eta }\) at the location \(\hat{y}=24,\) which reveals oscillations ranging from \(\widehat{A}=0.5\) to 0.56,  the latter at the initial time, and then more or less settling around \(\widehat{A}=0.52\pm 0.01.\) Minor waves dispersing from the main three solitons lead to an increase of this local maximum relative to the \(\tilde{A} = \frac{1}{2} (3/4)^{1/3}\approx 0.4543\) or \(\widehat{A}=0.5.\) Figure 8c reveals that the maximum amplification \(\max (\hat{\eta })/\widehat{A}\) over time reaches a maximum of circa 8.8,  which thereafter settles to circa 8.2. Finally, Fig. 8d concerns the energy evolution over time relative to the initial energy. The energy deviations fluctuate around \(10^{-4}.\) While initially a clear downward drift of energy is seen, when we simulate beyond time zero, the energy deviations turn upward, indicating that the oscillations settle on the longer time-scale of the “cnoidal” period of these soliton interactions, cf. the simulations in the accompanying supplementary movie for [13]. This monitoring of energy suggests that our simulations are stable.

To assess how an increase in wave amplitude affects these amplification results, we show a BLE simulation with \(\epsilon =0.05\) in the four-panel graphs of Fig. 9. The BLE simulation time interval is now \([-60,20],\) different from the previous case with \(\epsilon =0.01,\) because the changed values of \(\epsilon \) and \(\mu \) affect the time transformation, cf. (7), between BLE and KPE by a factor of circa 2.24,  which is consistent with the ratio of times when the amplifications start to grow towards their maximum. The results remain largely similar, but oscillations due to smaller scale wave dispersion increase; see also Fig. 10. The maximum amplification lies within (7.6, 8.8). Notably, the energy deviations indicate numerical stability and remain around \(10^{-4}\) except near the initial time at BLE time circa \(\hat{t}=-60.\) In both simulations, there seems an initial small adjustment of the energy near to or at the initial time. Discussion regarding the information gleaned from these simulations is deferred until Sect. 6.

Fig. 10
figure 10

Numerical three-soliton solution for the Benney–Luke equations (BLE-SP3) for \(\epsilon =0.05\) at times \(\widehat{t} =-20\) (left-hand side panels) and \(\widehat{t} = 0\) (right-hand side panels), respectively. Panels a and b show the surface elevation \(\widehat{\eta }\) which is seen to attain a global maximum at \(\widehat{t}=-20\). Panels c and d show that the height of \(\widehat{\eta }\) in the far-field \(\widehat{y} = 24\) is close to the initial height 0.5 at both times. Panels e and f show the potential \(\tilde{\Phi }\)

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5.2 Amplification in the Potential-Flow Equations

Simulations of the PFE with the travelling Y-shaped soliton-complex of two interacting solitons of equal amplitude successfully show that the fourfold amplification in that complex can persist at about 3.85 to 4.05,  albeit with some dispersive effects for \(\epsilon =0.05\) and \(A\approx 0.416\) (not shown here, see [28] and Appendix B).

Fig. 11
figure 11

Numerical results of \(\eta \) and \(\tilde{\psi }\) obtained from a potential-flow simulation of three-soliton interactions (PFE-SP3) with \(\epsilon =0.01.\) a–d used CG2 basis functions with spatial step \(\Delta x\approx \Delta y=200~\text {m}\) and time step \(\Delta t=0.7139~\text {s},\) and e–h are \(\eta \) at \(t=3426.8~\text {s}\) using different basis functions and spatial-temporal resolution as indicated in panel sub-captions

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Fig. 12
figure 12

Cross-sections of \(\eta \) from PFE-SP3 with \(\epsilon =0.01.\) All curves are close to each other which implies that the resolution seems to be converged. Panels a and b show that the height of \(\eta \) in the far-field \(y = 35{,}000~\text {m}\) is close to the initial height \(0.1~\text {m}\) at both times \(t=3426.8~\text {s}\) and \(t=6853.6~\text {s}.\) Panels c and d depict that the height of \(\eta \) at the bottom boundary \(y=0~\text {m},\) at time \(t=3426.8~\text {s}\) and \(t=6853.6~\text {s},\) respectively. In each case, the free surface attains a height of about \(0.82~\text {m}\) and \(0.89~\text {m},\) respectively

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Fig. 13
figure 13

Results obtained from a potential-flow simulation of three-soliton interactions (PFE-SP3) with varying spatial resolution. The simulation employs parameters in the table above. We set the base resolutions in space and time as \(\Delta x\approx \Delta y=200~\text {m},\) \(\Delta t=\Delta t_{\textrm{BLE}}(H_0/\sqrt{g H_0\epsilon ^2})\approx 0.005\times 142.78=0.7139~\text {s}\) where \(g=9.81~\text {m}/\text {s}^2.\) In addition, the initial far-field amplitude of each soliton was set to \(\tilde{A}=0.5\left( \frac{3}{4}\right) ^{1/3},\) so that \(A=0.1~\text {m}\) (cf. (2) and (11a). Key to colours: blue—CG2 basis functions with \(\frac{\Delta y}{3}\) and \(\Delta t;\) orange—CG4 with \(\frac{2\Delta y}{3}\) and \(\Delta t;\) green—CG2 with \(\frac{\Delta y}{3}\) and \(\frac{\Delta t}{2};\) red—CG2 with \(\frac{\Delta y}{4}\) and \(\Delta t\) (relative to base resolution). Note that the basis functions quoted above refer to the horizontal direction only; in the vertical direction, CG2 is always used. The purple and brown curves, respectively, denote rescaled BLE and KPE solutions, used as reference. Panels show: a maximum values of \(\eta \) over time; b \(A:=\max _x(\eta (\cdot ,y=35{,}000~\text {m},t));\) c amplification \(\max (\eta )/A\) against time; and d relative-energy error \(\vert 1-E(t)/E(0)|.\) All the initial total energies E(0) are the same, \(-2.782\times 10^{13}\) up to five digits. Run times given are for simulations using 40 cores on arc4 HPC. We note that the simulation on the second row requires a much longer run time than the first simulation, despite having the same DoFs, and this is caused by the CG4 basis functions that lead to denser operator matrices

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We therefore carry out simulations of the PFE using \(\epsilon =0.01,\) seeded with the KPE three-soliton solution well prior to the KPE time \(\tau =0\) of maximum amplification, i.e., they are seeded at a time when the maximum amplification of the three far-field solitons at \({A}= 0.1~\text {m}\) is found in two (local) maxima of 4A. The solution around these two local maxima consists essentially of the travelling Y-shaped soliton. While each Y-complex is steady, they are travelling in different directions and their interaction creates the time-dependent three-soliton with its maximum ninefold peak in space and time.

The ‘standard’ or base resolution used in the PFE simulations is determined as follows. We first choose the domain length and base mesh size in the y-direction as \(L_y=(H_0/\sqrt{\mu })L_{y{\textrm{BLE}}}=40{,}000~\text {m}\) and \(\Delta y=\Delta y_{\textrm{BLE}}(H_0/\sqrt{\mu })=200~\text {m},\) with \(L_{y{\textrm{BLE}}}=20\) and \(\Delta y_{\textrm{BLE}}=0.1\) (the remaining parameter values are given in the table of Fig. 13 caption). This results in a base number of elements in the y-direction equal to \(N_y=L_y/\Delta y=200.\) The base mesh size in the x-direction, \(\Delta x,\) is chosen to be similar to \(\Delta y\) by selecting a number of elements as the nearest integer, \(N_x,\) to \(L_x/\Delta y,\) where \(L_x=(H_0/\sqrt{\mu })L_{xBLE}=17{,}725.6~\text {m}\) and \(L_{xBLE}=8.863.\) Therefore, the base mesh uses a number of degrees-of-freedom (DoFs) equal to \((n_hN_x)(n_hN_y+1)(n_vN_z+1)=(2\times 89)(2\times 200+1)(2\times 4+1)=642,402,\) where \(n_h,\) \(n_v\) is the order of base functions in the horizontal and vertical, respectively (in all simulations, we used \(n_h=2\) or 4 and \(n_v=2;\) see also the discussion in the paragraph after Eq. (43). In addition, the base time step is \(\Delta t=\Delta t_{\textrm{BLE}}(H_0/\sqrt{g H_0\mu })=0.7139~\text {s},\) with \(\Delta t_{\textrm{BLE}}=0.005.\) Subsequently, we refine resolution to check convergence of the simulation undertaken with the lowest resolution presented, i.e., denoted by “CG2/\(\frac{\Delta y}{3}\)”. For spatial refinement, we take \(N_y=pL_y/\Delta y\) (with \(p>1\)) elements in the y-direction and appropriately choose a number of elements in the x-direction, \(N_x\) to retain the scaling \(\Delta x\approx \Delta y.\) The DoFs for all the refined simulations are given in the table of Fig. 13 caption.

Two groups of numerical results are shown in Fig. 11; snapshots (a)–(d) display the fields of surface elevation \(\eta \) and free-surface velocity potential \(\tilde{\psi }\) at the mid- and end-run times of a simulation using CG2 basis functions; and (e)–(h) demonstrate the surface elevation \(\eta \) for various spatial and time resolutions. In addition, cross-sections of \(\eta \) at \(y=35{,}000~\text {m}\) and \(y=0~\text {m}\) are depicted in Fig. 12. It can be observed from the similarity of the various curves displaying \(\eta \) that spatial convergence is achieved. Further results of the simulations and the exact KPE solution are displayed in the panels of Fig. 13 for various spatial resolutions, time steps, and basis functions: the maximum of \(\eta \) over time, the far-field estimate of amplitude A at \(y=35{,}000~\text {m}\) over time, the maximum amplification over time as the ratio thereof as well as the relative-energy deviation. The amplification oscillates between about 7.6 to 8.25 due to dispersive effects, possibly due to the weaker dispersion and stronger nonlinearity in the PFE. The relative-energy deviations are small and appear to decrease for the higher resolutions, which is indicative of a stable simulation. It should be noted that smaller spatial resolutions in y-directions are required to get convergent results; here, \(\Delta y/3\) is deemed to be sufficient.

5.3 Comparison of Simulations

Figure 13 also presents a comparison between simulations of the BLE and the PFE, both using \(\epsilon =0.01,\) \(\delta =10^{-5},\) \(ac=1\) (we note that in Fig. 8 for BLE, a smaller value \(\delta =10^{-10}\) was used). The comparison reveals different amplifications at (8.8, 9) and (7.6, 8.25) for BLE and PFE, respectively. The more intensified oscillations seen in the latter are thought to be due to the stronger nonlinearity and weaker dispersion in the PFE, relative to those in the BLE and the KPE. At present, PFE simulations for larger \(\epsilon =0.05\) and \(\widehat{A}=0.5\)—as performed for the BLE and shown in Fig. 9—are not attainable without wave-breaking parameterisations and/or a more advanced treatment of the numerical or exact KPE solution for periodic channel domains.

6 Concluding Remarks

In the present investigation, we generated extreme waves arising from interaction of three line-solitons with equal far-field amplitudes by seeding, with exact solutions of the Kadomtsev–Petviashvili equation (KPE), finite-element solvers based on two water-wave models, i.e., the Benney–Luke equations (BLE) and the more advanced potential-flow water-wave equations (PFE). We have additionally developed new analytical results on the theoretical ninefold maximum amplification of the KPE, and have moreover established that an amplification factor of circa (7.6, 9) for three interacting solitons can be reached in numerical simulations of the BLE and PFE.

We have carried out two sets of simulations using soliton wave amplitudes computed with \(\epsilon =0.01,0.05.\) To set up these simulations, we seeded them with an exact solution of the Kadomtsev–Petviashvili equation (KPE) at the time when the amplification is fourfold relative to the far-field soliton amplitude. An advanced analysis has been presented to prove that this KPE solution consists of three-soliton interactions that can reach a ninefold amplification at a later point in space and time, which discovery both removes a spatial divergence from our previous analysis [13] and also enables results to be computed with higher amplifications in our newest simulations. The simulations with \(\epsilon =0.05\) are reasonably realistic inasmuch as the far-field soliton amplitude is then \(0.5~\text {m}\) when the rest depth is taken as \(H_0=20~\text {m}\) under the scaling (2); these parameters yield a simulated peak wave of about 3.5 to \(4~\text {m},\) while the peak in the KPE would reach \(4.5^{-}~\text {m}.\)

To conduct the PFE simulations, we have employed a pseudo-computer-algebraic approach of directly implementing the relevant time-discrete variational principles (VP) within the finite-element framework Firedrake. That led to a spatially spectrally accurate water-wave model with second-order accuracy in time. The direct implementation of the time-discrete VPs reduced time-to-development while closely following the mathematics; additionally, the respective weak formulations of the discrete equations were generated automatically by Firedrake, thereby avoiding derivation of cumbersome expressions of the weak forms and hence circumventing the potential introduction of human error. Subsequent optimisation of the solver pre-conditioners for the nonlinear equations gave a speed-up factor in the interval [4, 9] on both laptops and HPC computers.

The simulation with the PFE revealed an amplification of \(\sim \!7.8\)—relative to those in the BLE of (8.8, 9)—for amplitude of \(A=0.1~\text {m}\) (both for \(\epsilon =0.01\)). Higher amplitudes are presently not (yet) achievable in the current channel-periodic domain and set-up, and/or without wave-breaking parameterisations. Despite our computational advances, it was difficult to achieve simulations of higher amplitude waves with acceptable stability, as evidenced by monitoring the temporal evolution of energy fluctuations that should be devoid of a systematic drift. The number of required degrees-of-freedom for the PFE simulations is large, and the narrow- and high-peak amplitudes demanded a level of resolution that has hampered matters. The difficulty of finding adequate resolutions may have been caused by the manner in which we incorporated the x-periodicity, since the background velocity-potential fields \(U_0(y,z)x\) used to achieve that periodicity for the variables representing the velocity-potential deviations created numerical sensitivity at the top corners of the channel.

One potential way to deal with the stronger nonlinearity experienced in the PFE is to include all higher order terms in (12) and (13), such that the initialisation is improved, given that we noticed a strong difference in dispersion between the KPE-BLE models and the PFE at higher values of \(\epsilon \) and \(\mu (=\epsilon ^2).\) Another way to avert this sensitivity at the top corners is to analyse exact or semi-analytical cnoidal-wave versions of the KPE solution on an infinite plane, and to use these to seed BLE and PFE simulations. However, these exact solutions could be more complicated [29] than the approach presently adopted and so it remains to be seen whether a proof of exact fourfold and ninefold amplifications can be derived for such two and three cnoidal-wave interactions.Footnote 4

Finally, further extensions of the present work will be directed towards establishing not only simulations and a design to support experimental validation, but also faster computational solvers through improved pre-conditioning and non-uniform mesh refinement.