# Phase Dynamics of the Dysthe Equation and the Bifurcation of Plane Waves

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## Abstract

The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such occurrences may be described using a Korteweg–de Vries equation. The solitary wave solutions of this system form a qualitative prototype for the bifurcating dynamics, and the role of mean flow and steepening is then made clear through how they enter the amplitude and width of these solitary waves. In addition, higher order phase dynamics are investigated, leading to increased nonlinear regimes which in turn have a more profound impact on how the plane waves transform under defects in the phase.

## Keywords

Modulation Phase dynamics Dark solitary waves Wave–mean flow coupling## 1 Introduction

At the heart of the modern study of waves is their behaviour and stability. The last century has heralded many studies and successes into these avenues, but there is much left to be understood. Particularly, the stability of monochromatic wavetrains in hydrodynamics generated large interest after the experiments of Benjamin and Feir demonstrated that such a state was unstable in experiments [4, 5]; some analytical insight has since been gained using various mathematical techniques [44]. Such instabilities have been speculated to lead to the formation of rogue waves [34, 43], or the decrease in the frequency of the monochromatic wave [7, 28, 41]. However, even when such waves are stable to an instability like this, they can undergo different transitions to generate structures such as dark solitary waves [24, 30, 32], whose mechanism for formation remains unclear. Therefore, even such a heavily studied problem has a wealth of dynamics we have yet to understand.

*A*(

*x*,

*t*), velocity potential \(\Phi (x,z,t)\) of the flow, and

*h*the depth of the fluid. The constants \(\alpha \) and \(\beta \) characterise the magnitude of mean-flow and higher order self-steepening effects, respectively. In the work of Trulsen and Dysthe [46], the values of these parameters are taken to be \(\alpha = 4\) and \(\beta = 8,\) but we shall leave these free to see their explicit role in the emergence and evolution of defects.

*k*and frequency \(\omega \). Then, the key idea is to assume that these wave variables are not fixed, but instead slowly vary in time. Under this assumption, either by an averaging principle or direct asymptotic analysis, one generates the system of equations:

*k*(

*X*,

*T*), frequency \(\omega (X,T)\), and slow variables \(X = {\varepsilon }x,\,T = {\varepsilon }t\) for \({\varepsilon }\ll 1\). The functions

*A*and

*B*turn out to be the wave action and wave action flux, respectively, for the original system, averaged over one period of the original wavetrain \(\hat{u}\). This set of equations then govern how the wavenumber and frequency evolve, which lead to deformations in the wavetrain from which they are derived. There have been several extensions to the Whitham methodology, such as the extension to problems with multiple phases [1, 35, 50] and to the more general setting of relative equilibria [10, 49, 50]. It is these extensions that will allow us to explore the phase dynamics of the Dysthe system (1), as one not only modulates the emergent wavetrain but also the mean flow, which itself forms a relative equilibrium.

The Whitham equation (2) has the additional usage that by investigating the linearised stability problem for a fixed wavenumber and frequency, one can infer stability properties of the original wave. This is diagnosed by the properties of the eigenvalues of such a linearisation, which are denoted as the characteristics of the system. For real characteristics, the system is hyperbolic and the associated wavetrain is stable. Alternatively, in cases, where the characteristics become complex the Whitham system is elliptic and the underlying wave is unstable, with perturbations to it growing exponentially. One of the most famous examples of this usage arises when studying the Whitham equations which emerge from the nonlinear Schrödinger equation. One can show in this case that the Benjamin–Feir instability criterion emerges at the point when the characteristics become complex [29]. This result was the moment when “the penny dropped” for Whitham regarding the connection between characteristics and stability [33].

In summary, this paper aims to utilise a modification of the Whitham modulation theory to derive nonlinear dispersive equations to investigate the phase dynamics of the Dysthe system (1). This will require a modulation of a two-parameter relative equilibrium, namely, the plane wave solution coupled to a uniform current, which adopts a theory adept at treating multiple phases as well as a moving frame. This is achieved by combining existing modulation approaches, and will ultimately be used to show how the KdV equation governing the phase dynamics emerges from the Dysthe equation. Moreover, a secondary aim will be to provide criteria for when one expects the behaviour of defects in the wave to be qualitatively different, occurring precisely when one of these reductions fails to be adequate and should be replaced by another phase dynamical equation. In the setting of this paper, we consider only one example of this, which will focus on how the Gardner, and consequently the mKdV equation, arise whenever the nonlinearity of the KdV equation is sufficiently small and vanishing. This leads to increased nonlinear effects within the evolution of phase defects, which will be highlighted in the body of the paper.

From this phase dynamical description, we aim to provide a possible qualitative picture for the emergence of various coherent structures, such as dark and bright solitary waves. In the study of this paper, we will utilise the various nonlinear dispersive reductions and illustrate how these structures can be interpreted as a deformation under the perturbation of the wave variables by the solitary wave solutions of these systems. This technique has been utilized in other works [3, 8, 23, 26]; however, it has yet to be considered in the context of a wave–mean flow system like the Dysthe equation. Thus, another contribution of this work will be to determine how this coupling manifests within this approach as well as to identify its effect on the qualitative predictions. Depending on the phase dynamical description, which itself depends on the properties of the moving frame, we will illustrate that various structures are predicted to form from this viewpoint. Moreover, we will demonstrate that each regime implies that the original wave is affected to differing degrees. Thus, the characteristic speeds from the Whitham equations may in fact be used as a diagnostic regarding the degree in which the phase dynamical description distorts the original wavetrain.

The outline of this paper is as follows. In Sect. 2, we review how the existing phase modulation approach may be applied to the Dysthe system (1) to obtain a KdV equation which governs how the phases of the waves evolve over space and time. This is followed in Sect. 2.1 by an illustration of how such a KdV may be utilised to predict the bifurcation behaviour of the original plane wave solution, with an emphasis on processes that lead to the formation of dark or bright solitary waves. Subsequently, in Sect. 3, further phase reductions are discussed, noting when such equations arise and their effect on the plane wave solution. Concluding remarks are presented in Sect. 4.

## 2 From Dysthe to KdV

There are a multitude of techniques in which to investigate the phase dynamics of equations like (1). Primarily, the approach usually taken is to use a standard multiple scales analysis and solve at each order of the small parameter [8, 26]. A variant of this, and the closest reduction procedure to the modulation considered here, is to undertake this approach after transforming the system using the Madelung transform [3, 17, 23]. This approach is utilised in Appendix B to derive the KdV and show that it agrees with that obtained via the modulation methodology. Prior to this paper, it does not appear that such a study using the Madelung approach has been undertaken in the context of wave–mean flow systems such as the Dysthe equation, and so the reduction using this method is also another novelty of this paper. Within this paper the focus will be on the modulation approach, with the details of how the KdV equation may be obtained using this procedure appearing in Appendix A. This method allows one to draw a connection between not only the conservation laws the Dysthe system possesses, but also highlights how the hyperbolicity of the system plays a role in the evolution of the phase defects.

*A*. This corresponds to the uniform wavetrain state in

*A*and the uniform flow solution of \(\Phi \), which explicitly is given by

*c*, from the solution (4) is given by (21) in Appendix A. For the Dysthe equation, this requires that the determinant of the Jacobian

*c*is real, we can define the relevant eigenvector of \(\mathbf{E}(c)\) required for the theory, \({\varvec{\zeta }}\), as

*c*and the self-steepening effects determined by \(\beta \) appear within the coefficients of the time and nonlinear terms. Thus, these effects have a non-negligible effect on the phase dynamics which emerge from the plane wave solution. The aim now will be to investigate how these effects influence the phase dynamics and lead to the emergence of coherent, localised structures from the original plane wave.

### 2.1 The Evolution of Phase Defects

*a*and width

*W*are given by

*U*, and thus the width of the observed disturbance to the wave, increases. This is the case for either choice of the characteristic speeds, as made clear by (6) which highlights that

*V*. Thus, the mean flow is the dominant effect in the width, as the steepening only asymptotically decreases the width as \(\beta ^{-\frac{1}{2}}\), whereas the mean flow causes this to grow as \(\alpha \).

The effect of the phase dynamics on the amplitude of the plane wave is less clear and involves an interplay between steepening and depth. However, by studying (10), one is able to see that overall the effects of steepening and the mean flow decrease the resulting amplitude of the structure which forms, and the asymptotic decay is algebraic of the order \(\alpha ^{-1},\,\beta ^{-1}\). As such, one expects the largest amplitudes for these disturbances to occur for \(\alpha = 0\) and for values of \(\beta \) approaching the singularity in the amplitude of the solution *U* as given in (8). However, in this proximity the KdV equation (7) begins to become an invalid model of the phase dynamics and other descriptions become operational, as will be described within the next section, so the discussion here will not apply to such choices of \(\beta \) that are sufficiently close to these points.

In summary, this approximate picture provides us with a qualitative description for how these localised structures appear and what form these are expected to take. In the presence of strong mean flow interactions between the wave and the current, the distortion to the envelope is expected to have a longer range. When the wave is subjected to increased self-steepening effects, the magnitude of the disturbance is muted so long as a degeneracy in the derived KdV equation is not approached in parameter space. Therefore, the KdV dynamics derived in this setting suggest that the original plane wave is in fact stable to the majority of defects which can form, and their effect on the wavetrain is very limited.

We now discuss the results of this investigation. One can observe that the mean flow and steepening effect can lead to qualitative differences in the dynamics, changing the polarity of the resulting structure from dark to bright, and viceversa. This is to be expected, since the steepening effects enter the nonlinear coefficient and allow for it to change sign. Moreover, we can see that these effects can also lead to the previously discussed suppression of the solitary structures, suggesting that the original wave persists even in the presence of the nonlinear phase dynamics and hints at increased stability in these regimes. Additionally, it becomes clear that the presence of mean flow and self-steepening within the system can lead to increased nonlinear effects in the predicted bifurcating behaviour, where these disturbances are enhanced. Overall, the presence of both of these effects leads to a greater range of behaviours for the localised structures.

## 3 Further Singularities and Higher Order Phase Dynamics

Although the KdV itself provides some insight into the evolution of defects and the formation of coherent structures from the uniform wavetrain, there are parameter values for which the KdV stops being operational due to vanishing terms. At such points, there is the potential for even more interesting nonlinear phenomena to emerge in a way analogous to a secondary instability via higher order phase equations. We discuss one such case of this below, leading to the modified KdV equation, and by extension the Gardner equation, with a discussion of their effects on the evolution of the original plane wave solution.

### 3.1 Modified KdV

These observations are confirmed by the examples of how these solutions modify the original plane wave, depicted in Fig. 2. Overall, the reconstructions in this case tend to have a more pronounced but typically broader effect on the plane wave than the KdV equation, causing more apparent versions of the emergent dark and bright solitary waves. This is expected, as the mKdV soliton is both narrower and larger in amplitude than the one which is admitted by the KdV. Moreover, the scalings within the ansatz to obtain the mKdV are larger than the KdV case. Thus, the resulting bifurcation that the phase dynamics predicts should be of higher magnitude, as well as being sharper in regimes, where the mKdV (13) is operational.

### 3.2 Gardner Equation

*B*is large is reminiscent of the mKdV case, with more pronounced bifurcating structures than for the KdV. For lower

*B*(corresponding to larger \(\gamma \)) one observes slightly wider packets of lower amplitude, closer to those of the KdV dynamics. Thus, the Gardner equation sheds light on the dynamics in the intermediate regime between the KdV and mKdV equations, and thus the way in which the solitary waves become tighter and increase in amplitude, as one may expect.

## 4 Concluding Remarks

This paper has demonstrated that there is a great wealth of phase dynamics emerging from the Dysthe equation, primarily owing to both mean flow and self-steepening effects. The simplest of these is the KdV, which predicts how plane waves can bifurcate to modulated, or dark/bright structures. Additionally, when certain criterion are met, richer dynamics can occur by the increment of nonlinearity within the phase equations.

Although the phase dynamics provides a qualitative picture as to the formation of solitary structures within the original plane wave, the next step would be to compare the results obtained here to those from direct numerical simulation. This would quantify the ability of the nonlinear dynamics discussed here to capture the true bifurcating behaviour, and possibly help to lend these analytic techniques more credibility when discussing these scenarios.

There are other systems which exhibit a mean flow coupling within a set of nonlinear equations with a free surface. Examples include a version of (1) with higher order terms in *A* or systems such as Benney–Roskes [6] or Hasimoto–Ono [21] equations. These can also be explored using a phase dynamical approach to investigate how the flow beneath a wave influences how the uniform wave is modulated in the presence of defects. The novel features of the latter systems is that the wave action vectors these problems admit are nondegenerate, unlike the Dysthe equation, and so richer time dynamics are possible.

## Notes

### Acknowledgements

The author would like to thank helpful discussions and feedback from Karsten Trulson, Miguel Onorato and Ying Huang during the formulation and writing of this paper.

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