Nonlinear theory for coalescing characteristics in multiphase Whitham modulation theory

The multiphase Whitham modulation equations with $N$ phases have $2N$ characteristics which may be of hyperbolic or elliptic type. In this paper a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation. That is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling-wave solutions of coupled nonlinear Schr\"odinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.


Introduction
The theory of modulation, particularly Whitham modulation theory, takes existing nonlinear waves, such as finite-amplitude periodic travelling waves, and provides a framework for studying the dynamical implications of perturbing the basic properties of the nonlinear wave. In classical modulation, the properties of the basic state (wavenumber, frequency, meanflow) are allowed to depend on space and time, and partial differential equations (PDEs) are derived for these parameters. Study of these PDEs then provides information about the evolution of the basic state under perturbation.
Given a basic state, there are several strategies for deriving modulation PDEs (averaging the Lagrangian, averaging conservation laws, geometric optics ansatz, other ansätze). In all cases the governing equations produced by Whitham modulation theory, for a simple one-phase periodic travelling wave, can be expressed in the canonical form q T = Ω X and ∂ ∂T A (ω + Ω, k + q) + ∂ ∂X B(ω + Ω, k + q) = 0 . (1.1) They are a pair of nonlinear first-order PDEs for the two unknowns Ω(X, T ), the modulation frequency, and q(X, T ), the modulation wavenumber. The parameters (ω, k) are representative of the wavetrain from which the Whitham modulation equations are obtained, and X = εx and T = εt are slow time and space scales. The first equation is called conservation of waves and the second is called conservation of wave action [43]. When the governing equations are the Euler-Lagrange equations associated with a Lagrangian functional, the scalar-valued functions A and B are related via The function L (ω, k) is obtained by averaging the Lagrangian evaluated on the periodic travelling wave with frequency ω and wavenumber k.
The pair of quasilinear first-order equations (1.1) can be classified based on their characteristics. The Whitham modulation equations (WMEs) can either be hyperbolic (real characteristics) or elliptic (complex characteristics) and the transition signals a change of stability of the underlying periodic waves [41,43,9,10]. It is this change of type, and its generalization to multiphase wavetrains, and its nonlinear implications, that are the main themes of this paper.
To identify the structure of coalescing characteristics, first consider the one-phase case where only two characteristics exist and so coalescence is elementary. The linearization of the one-phase WMEs (1.1) about the basic state, represented by (ω, k), is q T = Ω X and A ω Ω T + A k q T + B ω Ω X + B k q X = 0 , (1.3) or, under the assumption A ω = 0, they can be written in the standard hydrodynamical form, (1.5) Here, A and B are evaluated at Ω = q = 0. The characteristics (eigenvalues of F) are where using (1.2) in the latter equality. The sign of the determinant ∆ L , called the Lighthill determinant (Lighthill [23]), signals whether the characteristics are real or complex, At the transition, when ∆ L = 0, the two characteristics are equal, Whitham modulation theory breaks down, and a new modulation strategy is needed. In [9] a nonlinear modulation theory is developed for breakdown of the WMEs in the case of one-phase wavetrains. It is valid near the transition from hyperbolic to elliptic, showing that the WMEs (1.1) are replaced by q T = Ω X and A ω Ω T + κqq X + K q XXX = 0 , (1.8) where T = ε 2 t, X = ε(x − c g t), and c g is a nonlinear group velocity at the transition. The coefficients A ω and κ are obtained from derivatives of the components of conservation of wave action, and the dispersion coefficient K arises due to a symplectic Jordan chain argument. Differentiating the second equation of (1.8) with respect to X and using the first equation reveals that it is a variant of the two-way Boussinesq equation for q, A ω q T T + 1 2 κq 2 + K q XX XX = 0 , (1.9) The coefficients in (1.8) and (1.9) are universal in the same sense that the Whitham equations are universal -they follow from abstract properties of a Lagrangian. Extension of the derivation of (1.8) to two space dimensions and time appears in [10]. The emergence of the equation (1.9) shows that coalescing characteristics generate nonlinearity, dispersion and wave fields of greater complexity. The complexity is due to the wide range of known localized, multi-pulse, quasiperiodic, and extreme value solutions of the two-way Boussinesq equation.
In order the generalize this nonlinear theory for coalescing characteristics to the case of multiphase wavetrains several new results are needed. In the case of one-phase wavetrains there are only two characteristics and so the coalescence can only happen in one way. In the multiphase case with N −phases there are 2N characteristics and a more sophisticated theory is needed to identify coalescing characteristics and assure that they change type. In addition, several facets of the linear theory, such as intertwining Jordan chains, that generate the coefficient K , bring in new challenges. The third major generalization needed for coalescing characteristics is a new nonlinear theory. We will find that the form of the twoway Boussinesq equation (1.9) carries over to the multiphase case but there is a discrepancy between the fact that (1.9) is scalar valued but the WMEs in the multiphase case have 2N equations. Hence a secondary reduction of the nonlinear equations will be required. Showing that the coefficients are universal is also an order of magnitude more difficult in the multiphase case.
The problem of how characteristics coalesce and change type is addressed as follows. Firstly consider the one-phase case. The change of type of the characteristics signals an instability of the basic state, and this linear instability is made apparent by taking the normal-mode ansatz q(X, T ) Ω(X, T ) = Re q Ω e λT +iνX , and substituting into (1.3) to obtain λ = ic ± ν .
An unstable exponent (Re(λ) > 0) with modulation wave number ν exists precisely when ∆ L > 0. As ∆ L changes sign the eigenvalues change from purely imaginary to a complex quartet as shown schematically in Figure 1. This type of stability transition is familiar from the theory of linear Hamiltonian systems, as it is precisely the Hamiltonian Hopf bifurcation [40], and in that setting the collision and resulting instability occurs since the eigenvalues have opposite Krein signature [20,40]. However, as shown in [11], there is no obvious symplectic structure in the Whitham theory, and it is the sign characteristic of Hermitian matrix pencils that is operational here. The sign characteristic has a central role in the theory of Hermitian matrix pencils relative to an indefinite metric (see Gohberg et al. [17] for a history and references). The Hermitian matrix pencil structure of (1.3) is evoked by multiplying the conservation of waves by A ω , assuming A ω = 0, and combining the two equations in (1.3) as The two coefficient matrices are symmetric. Now the modified normal mode ansatz generates the following Hermitian matrix eigenvalue problem The theory of Hermitian matrix pencils shows that each eigenvalue of (1.11) has a sign characteristic and a necessary condition for instability is that eigenvalues coalesce and have opposite sign characteristic [11]. In the one phase case, with just two characteristics, the sign characteristic is less interesting, and indeed trivial. In the multiphase case, with many characteristics, the coalescence of characteristics may or may not lead to instability, and the sign characteristic becomes an essential diagnostic tool. The principal case of interest in this paper is when all the characteristics are real, with only one pair, having opposite sign, undergoing a transition to instability. The generalization of the dispersionless WMEs (1.1) to the multiphase case is where ω, k ∈ R N are given parameters representative of the basic state, and q, Ω ∈ R N are the vector-valued unknowns, the modulation wavenumber and frequency, which depend on T = εt and X = εx. When the governing equations are the Euler-Lagrange equation associated with a Lagrangian functional, the mappings A and B are variational with the properties, and (1.14) Cross differentiating shows that the Jacobians satisfy This symmetry will be important for generalising the Hermitian property of (1.11) to the multiphase case. Given a smooth function L , the pair of equations (1.12) is a closed first-order system of PDEs for Ω and q with 2N characteristics. This formulation of vector-valued WMEs was introduced in Ratliff [30,28]. However, multiphase Whitham modulation theory has a rich history. Multiphase WMEs were first introduced and studied by Ablowitz & Benney [1], in the context of scalar nonlinear wave equations, where the appearance of small divisors was noted. For integrable systems small divisors disappear: multiphase averaging and the Whitham equations are robust and rigorous, and a general theory can be obtained (e.g. Flashka et al. [15] and its citation trail). Whitham [43, §14.7] includes potential variables as additional phases ("pseudo-phases") and generates a form of multiphase modulation and applies it to wave-meanflow interaction of Stokes water waves [42]. Willebrand [44] takes multiphase modulation theory to a new level by deriving the N −phase WMEs, for Stokes wave solutions of the water wave problem, with N arbitrary and even considers the limit N → ∞. This theory is formal and the series are divergent and have small divisors, but the leading order terms are instructive (see comments on this later in §9). The theory of [44] is now used in ocean wave forecasting (e.g. Chapter 9 of Olbers et al. [27]).
On the other hand, when the system is not integrable, but there is an N −fold symmetry, a theory for conservation of wave action can be developed without small divisors and smoothly varying N −phase wavetrains. This strategy is implemented in [30,28], where multiphase wavetrains are characterized as relative equilibria with smooth dependence on parameters.
Going back to the abstract multiphase WMEs (1.12), with the symmetry property (1.15) and the gradient properties (1.13), the linearization of (1.12) can be cast into the form of a Hermitian matrix pencil, assuming that D ω A is invertible. The Jacobians D ω A, D k A, D ω B, and D k B, are N × N matrices, with the first two symmetric and the latter two related by transpose. The 2N ×2N linear eigenvalue problem (1.16) can be reduced, by eliminating Ω, 17) to an N × N quadratic Hermitian matrix pencil, A parallel theory can be developed for the sign characteristic in this context [16,26,38]. Suppose c 0 is a simple real eigenvalue satisfying det[E(c 0 )] = 0 with eigenvector Then the sign characteristic of c 0 is 20) where ·, · is an inner product on R N , and the prime denotes differentiation with respect to c. A discussion of the history and various formulations of the sign characteristic is given in [11]. The sign characteristic is invariant under congruence transformation, E(c 0 ) → P T E(c 0 )P, for any invertible P [16]. The quadratic formulation (1.18), rather than its linearisation (1.16), turns out to be the most efficient in applications and arises naturally in the modulation theory.
Starting with the quadratic Hermitian matrix pencil (1.18) a theory for the sign characteristic in the context of Whitham modulation theory is developed by Bridges & Ratliff [11]. The 2N characteristics of the linearized problem satisfy (1.21) Double non-semisimple characteristics, which characterize coalescence, and define the nonlinear group velocity c g , satisfy ∆(c g ) = ∆ (c g ) = 0 but ∆ (c g ) = 0 , with a single geometric eigenvector E(c g )ζ = 0 , (1.23) and generalized eigenvector (1.24) Solvability of (1.24) defines c g . All these properties follow from the structure of the linear operator E(c) with c ∈ R and are studied in [11] and the details required here are developed in §4 and §6. When multiphase modulation is introduced for the nonlinear problem, with an appropriate scaling, the vector-valued conservation of wave action (1.12) will be morphed into another form with dispersion. It will however still have dimension N , so a further reduction is necessary in order to obtain a generalisation of (1.9). The strategy is to split R N = span{ζ} ⊕ R N −1 . The geometric eigenvector ζ, defined in (1.19), provides a preferred direction in q−wavenumber space associated with the coalescence. This preferred direction is an essential part of the nonlinear modulation theory. It provides a projection operator so that the vector-valued conservation of wave action (1.12) can be reduced to a scalar equation, and this scalar equation, which also requires a rescaling of the slow variables, and extension of the analysis to fifth order in ε, is a geometric form of the scalar-valued two-way Boussinesq equation where µ and κ are determined by the geometry of the averaged Lagrangian L (ω, k) and K is determined by a twisted Jordan chain argument. The geometry of L (ω, k) is discussed in §2.2. The most remarkable outcome of the geometry is that the coefficient κ in (1.25) has the simple formula . (1.26) The coefficient µ is determined by a Jordan chain associated with the linear operator E(c g ) in (1.23). Indeed, µ = 0 is the condition required for termination of the Jordan chain (ζ, γ) in (1.23)-(1.24). A different Jordan chain, associated with the linearization of the Euler-Lagrange equations (denoted L in (2.9)), determines the dispersion coefficient K . This latter Jordan chain argument is similar to the case of multiphase modulation associated with zero characteristics in [30,28,33] but here the Jordan chain intertwines two different chains generated by L. The nonlinear modulation theory where (1.26) arises naturally and feeds into the emergence of (1.25) is developed in §5. The starting point for the theory is a general class of nonlinear PDEs generated by a Lagrangian, and this class is introduced in §2. Given a basic multiphase wavetrain Z(θ, ω, k), with phase θ = kx + ωt + θ 0 and vector-valued frequency and wavenumber ω, k, satisfying the Euler-Lagrange equations, the dispersionless vector-valued WMEs (1.12) are derived by modulating the basic state with a geometric optics scaling [30,28,11]. The appropriate modulation ansatz is Z(x, t) = Z(θ + ε −1 φ, ω + Ω, k + q) + εW (θ + ε −1 φ, X, T, ε) , (1.27) where φ, ω and q, depending on T = εt and X = εx, are the modulated phase, frequency and wavenumber, and W is a remainder. Substitution of (1.27) into the Euler-Lagrange equation and solvability requires q and Ω to satisfy (1.12) to leading order [30,28]. When two characteristics, of opposite sign characteristic, coalesce and transition to instability, the geometric optics modulation ansatz (with T = εt and X = εx) in (1.27) must be replaced. The altered form utilised is where Φ is a function of the slow time and space variables, with c g determined as part of the analysis, and ε measuring the distance in (ω, k)−space from the singularity (1.22). The new ansatz at coalescence is Finer detail on the ansatz, including definitions of q and Ω and their relation to Φ X and Φ T is given in §5. Substitution of this ansatz into the governing Euler-Lagrange equations, expanding everything in powers of ε, and setting order by order to zero, results, by imposing a solvability condition, in a vector-valued two-way Boussinesq equation induced by conservation of wave action. The projection operator, defined using Ker(E(c g )), is then implemented to split the conservation of wave action into two parts, one generating the two-way Boussinesq equation (1.25) with the complementary part carrying over to higher order. The theory applies to multiphase wavetrains with any finite-number of phases. But to avoid lengthy formulas, it is developed for the case of two-phase wavetrains. Two-phase wavetrains contain all the essential features of the multiphase case with coalescing characteristics. The results naturally extend to the N −fold case, so long as the eigenvalue conditions (1.22) and (1.23) and eigenvector structural requirements are satisfied.
The paper has four parts: the Lagrangian (geometry, analysis, and Euler-Lagrange equation), the linear theory (for both operators E(c) and L), the nonlinear modulation analysis (implementing the ansatz (1.30)), and an illustrative example.
In §2 a class of Lagrangian functionals and a class of basic states is introduced. In §2.2 the geometry of the mapping (ω, k) → L (ω, k), where L (ω, k) is the Lagrangian evaluated on a basic state, is studied. Remarkably, many of the features of the linear problem as well as the nonlinear modulation are determined by the geometry of this scalar-valued function. We end this discussion by reviewing the Whitham modulation theory from a geometric perspective in §3 to demonstrate how the characteristics and their coalescence may be formulated using these notions, as discussed in §4.
The linear theory has two parts: the structure of the linear operator E(c) in (1.18), including the Jordan chain theory in the setting of the quadratic eigenvalue problem (1.18). This theory is developed in §2.2 and §6, and appeals to the theory of sign characteristic for Hermitian matrix pencils developed in [11]. The second part of the linear theory is the linearization of the Euler-Lagrange equation, which is needed to develop a secondary Jordan chain needed for constructing the dispersion coefficient K and the nonlinear modulation theory, and this theory is developed in §6.
The nonlinear theory is developed in §5. Although the ansatz (1.30) with (1.28) is new, once the ansatz is identified the strategy is similar to our previous papers, particularly [9] and [33], and so only the key new features are highlighted. The theory is illustrated by application to the two-phase travelling wave solutions of a class of coupled nonlinear Schrödinger (CNLS) equations. In [11] it was shown that these travelling wave solutions have coalescing characteristics with transition to instability. Here the theory is applied to show the emergence of a geometric two-way Boussinesq equation at these singularities. Potential generalizations are discussed in the concluding remarks section.

The Lagrangian and governing equations
The theory is built on a general class of Lagrangian functionals The lower dots indicate that the Lagrangian may also depend on higher derivatives of V , and the subsequent theory can be adapted for these cases. Normally a non-degeneracy condition on derivatives of L with respect to V t and V x is assumed, but here these conditions are circumvented by assuming up front that the Lagrangian has been transformed to standard multisymplectic form where Z ∈ R n , M and J are skew-symmetric matrices, S : R n → R is a given smooth function, and ·, · is a standard inner product on R n . For definiteness, n is taken to be even and Examples with n = 4 for M and J include the dispersive shallow water equations [4,5], and the coupled-mode equation (also massive-Thirring equation), with an appropriate choice of S(Z) in both cases. Both cases satisfy (2.2) for all c ∈ R. These examples and others can be found in [6,7,5,4].
The Euler-Lagrange equation associated with (2.1) is The theoretical developments to follow are based on this abstract form of the Euler-Lagrange equation, with n even, and M, J general skew-symmetric matrices satisfying (2.2).

Symmetry, relative equilibria, and the basic state
The easiest way to generate smooth families of multiphase wavetrains is to consider a Lagrangian that is invariant under the action of a Lie group. Here, and henceforth it is assumed that the Lie group is two-dimensional and abelian. The principal examples are the two-torus T 2 = S 1 × S 1 , the cylinder S 1 × R, and translations in the plane R × R. The generalization to abelian Lie groups of any finite dimension is straightforward in principle but results in a proliferation of formulas and index notation. The theory will be developed for the case of the two-torus, which is appropriate for periodic two-phase wavetrains, as the translation group is much simpler and the necessary changes will be recorded when needed.
Assume that (2.3) is equivariant with respect to a two-torus, T 2 = S 1 × S 1 , with matrix representation G θ (an n×n orthogonal matrix) and θ = (θ 1 , θ 2 ). The infinitesimal generators are Since G θ is orthogonal the action of g j on Z is a skew-symmetric matrix. Equivariance of (2.3) then follows from the requirements The basic state, namely the solution which will be modulated, is taken to be a family of two-phase wavetrains of the form with θ (0) ∈ R 2 a constant, and This wavetrain is a multiparameter family of relative equilibria. Substitution of Z into (2.3) admits the governing equation for the two-phase wavetrain In the absence of symmetry, solutions of this problem may encounter small divisors. The advantage of the T 2 −symmetry of (2.3) is that multiphase wavetrains are smooth functions with no small divisors. The relative equilibrium structure of the basic state (2.6) then gives and can be thought of as the reference point along the group orbit.

Linearization about a multiphase wavetrain
Associated with (2.7) is the linear operator This operator is formally self adjoint with respect to the inner product Differentiation of (2.7) with respect to each θ i and each of the four parameters k i , ω i leads to the equations (2.11) The first of these results highlights the fact that the kernel of L is at least two dimensional, and in this paper it is assumed no larger, so that The other equations in (2.11) will become significant when the Jordan chain theory in a moving frame is developed. The assumption (2.12) along with the formal self-adjointness of L give the solvability conditions for an expression F to lie within the range of L as

Multisymplectic Noether theory
In the Lagrangian setting, the symmetry induces conservation laws via Noether theory. Transforming to a multisymplectic formulation then induces multisymplectic Noether theory which relates the structure operators J and M to the components of the induced conservation laws. Although these conservation laws may have other physical significance they play the role of conservation of wave action in the Whitham theory and so the components will be called wave action and wave action flux. There is a conservation law associated with each phase of the wavetrain, and multisymplectic Noether theory implies the existence of functions A j , B j satisfying and so where Z(x, t, θ 1 , θ 2 ) is a function of (x, t) and the phases (θ 1 , θ 2 ) which are here interpreted as ensemble parameters. Direct calculation verifies that the conservation laws are The components of the conservation laws can also be deduced directly from the averaged Lagrangian. The Lagrangian (2.1) evaluated on the two-phase wavetrain and averaged, is The wave action vector evaluated on the wavetrain is (2.17) and the wave action flux vector is By definition, we have the following The entries of these tensors are related to solutions via The definition of the wave action and wave action flux in terms of derivatives of the averaged Lagrangian induces symmetry of the Jacobians, The key property in both (2.17) and (2.18) is that the left-hand side is in terms of the functions of (ω, k) only and the right-hand side is expressed in terms of the properties of the Euler-Lagrange equation (2.3), namely through the structure matrices J and M. It is this connection that is the essence of multisymplectic Noether theory, and it feeds into the nonlinear modulation theory.

Geometry of the averaged Lagrangian
Many of the properties needed in the modulation theory can be deduced from the abstract mapping where L : R N × R N → R is the averaged Lagrangian (2.16) and is assumed to be a smooth function.
The wave action and wave action flux emerge from L via where ·, · is an inner product on R 2 (or R N if N −phases are operational). The second derivative can be used to generate the linear operator E(c). First set u = cv in the above expression and look at the derivative Hence The most remarkable result following from derivatives of L is the expression for κ, the coefficient of nonlinearity in the emergent two-way Boussinesq equation (1.25). Introduce the one parameter path in L (ω, k), with c g here considered as fixed, and ζ ∈ Ker(E(c g )). Then differentiating and using (1.13) and (1.14) gives However, it is the third derivative of F (s) that is of most interest. The formula for F (s) suggests that F (s) is a derivative of a path through the linear operator E(c g ), considered as a function of (ω, k) with c g fixed. Differentiating, (2.24) At this point, this expression is just a formula, but the inner product in the second row will emerge naturally in the modulation theory in a solvability condition, giving it relevance as the coefficient of the nonlinear term in the emergent modulation equation.
In a similar way, the coefficient µ in (1.25) can also be represented in terms of derivatives of L as in However, a more interesting characterization of µ is as a termination condition for the Jordan chain (ζ, γ) in (1.23)-(1.24) (see equations (4.9) and (4.10) below).

Generic multiphase Whitham equations
In this section a construction of the generic (distinct characteristics) multiphase WMEs is sketched from the paper's geometric perspective. It serves as a touchstone for the modifications needed for the non-generic (coalescing characteristics) case, and the generic theory is needed to define c g , the frame speed at coalescence. Attention is restricted to the case of two phases; the N −phase case follows similar lines. Given the basic state Z in (2.6), the generic WMEs are obtained using the geometric optics ansatz [30,28], with X = εx and T = εt, and the vectors φ, Ω, and q depending on X, T and satisfying conservation of waves q T = Ω X . Expand all terms in a Taylor series, e.g. W = W 1 + O(ε), substitute into (2.3) and solve the equations at each order of ε. At zeroth order the governing equations for the basic wave Z are recovered and at first order an equation for W 1 is obtained Applying the solvability conditions (2.13), and using the connection between the resulting expressions and the components of the conservation law (2.19a)-(2.19b), e.g.
then gives the generic WMEs, Taking into account that Z is a function of k + q and ω + Ω, averaging over the phase eliminates the ε −1 φ terms, and using the vector definition of wave action (2.17) and wave action flux (2.18), these two equations are the vector conservation equation which, when combined with conservation of waves and the symmetry condition give the generic WMEs in vector form. Further details of the above derivation can be found in [30,28]. A proof of validity of these multiphase WMEs, when the original equation is coupled NLS, covering both the cases of elliptic and hyperbolic characteristics, is given in Bridges et al. [8].
Consider the linearisation of (3.2) and (3.3) at (ω, k) Characteristics about any state (ω + Ω, k + q) can be obtained the same way, but here the main interest is in characteristics in the neighbourhood of the basic state. Differentiating the first equation and using the second results in a second order equation for q, With the normal mode ansatz A key property that we will need is that a simple root, say c 0 , has a "sign characteristic". A necessary condition for two characteristics to coalesce and transition from hyperbolic to elliptic is that they have opposite sign characteristic. A study of the sign characteristic in the context of the linearised multiphase WMEs is given in [11], and a sketch of this theory is given in the next section.  When there are N −phases this polynomial has degree 2N . The linear algebra of quadratic Hermitian matrix pencils can be found in [16,26,38] and references therein. Here a theory for the sign characteristic of simple roots and the theory of double non-semisimple roots is required.

Defining characteristics and coalescence
In the case N = 2 there are four characteristics and the case where two roots of (4.1) coalesce is of interest. It is not essential to the nonlinear theory for the other two characteristics to be hyperbolic at coalescence, although if they are not hyperbolic then the basic state is already unstable. For definiteness we assume that all the characteristics are hyperbolic and one pair transitions from hyperbolic to elliptic at some parameter value.
A characteristic is double when ∆(c g ) = ∆ (c g ) = 0 and ∆ (c g ) = 0 , where ∆(c) is defined in (4.1). The value of c at the collision is denoted by c = c g in anticipation of the connection with the concept of group velocity. The conditions (4.2) tell us that the algebraic multiplicity of c g is two. For Hermitian matrices the geometric multiplicity would also be two. However, Hermitian matrix pencils, in the indefinite case, can have non-trivial Jordan chains [17]. This property also carries over to Hermitian quadratic matrix polynomials [16]. Here we are interested in the case where the geometric multiplicity of E(c g ) is one To establish a Jordan chain, first look at the condition ∆ (c g ) = 0 in terms of the properties of E(c), where E(c) # is the adjugate [24]. Now use the fact that E(c g ) has rank one and the nonzero eigenvalue is Tr(E(c g )), This formula can be verified by direct calculation (see also [24]). Then and so with the assumption (4.3), Now look at this condition from the viewpoint of solvability, as that is how it will arise in the nonlinear modulation theory. In the case of algebraic multiplicity two and geometric multiplicity one, a Jordan chain for a quadratic Hermitian matrix polynomial has the form for some γ ∈ R 2 , if it exists [16]. Since E(c g ) is Hermitian (in this case real and symmetric), the solvability condition is ζ, E (c g )ζ = 0 confirming (4.4). Writing out this condition, gives a defining equation for c g Noting that D ω B = (D k A) T , this formula simplifies to The symbol c g is used as the derivative with respect to k over a derivative with respect to ω is reminiscent of the classical definition of group velocity. Termination of the chain (4.5) at length two is assured if the following equation is not solvable; that is where (4.5) has been used. This expression is called µ as another remarkable result in the nonlinear theory is that this coefficient is precisely the µ that appears as the coefficient of U T T in the emergent two-way Boussinesq equation (1.25). This connection will emerge in the nonlinear modulation theory. Further properties of this Jordan chain, and the Jordan chains associated with the linear operator L are discussed in more detail in §6, after the nonlinear modulation theory is introduced.

Nonlinear modulation at coalescence
For the nonlinear modulation near coalescing characteristics, the strategy is to introduce the ansatz (1.30), substitute into the Euler-Lagrange equation (2.3), expand everything in powers of ε, and set terms proportional to each order in ε to zero. The key step here is identifying the form of the ansatz. The role of frame speed is inspired by the one-phase case in [9], and the role of additional phase functions ψ and δ is inspired by [30]. These are included in the ansatz because they eliminate the need for functions that would appear from homogeneous solutions at each order. The proposed phase modulation is Then with Ω := φ T and q := φ X ⇒ q T − Ω X = 0 , and where θ, φ, ψ, δ, q, and Ω are all functions of X and T defined in (5.3), and c g is defined in (4.7). For ease, expand W in an asymptotic series, The remainder W could be defined as W (θ + εφ + ε 2 ψ + ε 3 δ, X, T, ε), to synchronise with the form of the modulation of the basic state, but is equivalent to the above formulation: expansion of W in a Taylor series in ε just changes the form of W j at each order, but the overall expansion gives equivalent results. Although the ansatz (5.4) is new, the expansion and substitution strategy is similar to our previous papers on multiphase modulation [33,28,30] and the single phase coalescing characteristics [9] and so only the key new points are highlighted. For example, at ε 0 order the governing equation for Z in (2.7) is recovered. At ε 1 and ε 2 order the generic 2−term Jordan chain in (2.11) is recovered as in the preceding works.
At third order in ε, after simplification, the system is where and c g is defined in (4.7). Applying the solvability condition (2.13) to (5.5) gives or, after using the conversions from the structure operators J, M to the functionals A j , B j in (2.19a)-(2.19b), the solvability condition can be written in the illuminating vector form Hence for solvability of (5.5) it is required that q X is in the kernel of E(c g ), for some scalar-valued function U (X, T ). It can be confirmed a posteriori that a(T ) does not contribute to the leading order result and can be neglected. Hence It is this scalar-valued function U (X, T ) that will ultimately be found to be governed by the two-way Boussinesq equation (1.25).
With the solvability condition satisfied, and the expression for q in (5.8), the complete solution at third order is An arbitrary amount of homogeneous solution can be added to W 3 but it is already incorporated into the functions δ and ψ in the ansatz. The equation Lv 3 = Kv 2 foreshadows a Jordan chain theory. The beginnings of the chain are in (2.11) which can be re-written as Lv 1 = 0 and Lv 2 = Kv 1 . This Jordan chain theory is developed in Section 6.

Fourth order
After simplification, the equation at fourth order is (5.10) The first inhomogeneous term feeds into the Jordan chain argument as it is of the form Lv 4 = Kv 3 , for some v 4 . For the other two inhomogeneous terms, apply the solvability conditions (2.13), and use the identities (2.19a)-(2.19b), to obtain This equation is of the form (4.5); that is, the Jordan chain associated with E(c). The theory of this Jordan chain is developed below in §6.2. Here, it is sufficient to use the argument presented in (4.5) and (4.7) for the chain (ζ, γ) of E(c g ). Applying that theory gives ψ XX = γU T (mod Ker(E(c g ))) , where "mod" signifies that an arbitrary amount of homogeneous solution can be included. This homogeneous solution can be neglected as it does not enter at fifth order. Thus the solution at fourth order is Fortunately this equation does not need to be solved explicitly. It feeds into the fifth order solution, and ultimately generates formulae for the coefficients, but these formulae will be obtained without an explicit expression for Ξ.

Fifth order
At fifth order, after combining terms and simplifying, the equations are The tilde above W 5 term indicates that the preimage of all terms lying in the range of L from the right-hand side have been absorbed (e.g. terms that would vanish identically under the solvability conditions). These terms come into play only at higher order. It is the solvability condition for this fifth order equation that will deliver the modulation equation for U (X, T ). However, solvability is a multistage process. There are two solvability conditions associated with the operator L (and N when there are N −phases), leading to a vector-valued equation. A secondary solvability condition, associated with the operator E(c g ), will reduce vector equation to the scalar two-way Boussinesq equation.
First establish the vector solvability condition. Apply the L−solvability (2.13) condition to the right-hand side of (5.15) term by term. Solvability of the U XXX term generates the vector Z θ 1 , Kv 4 Z θ 2 , Kv 4 U XXX := −TU XXX . (5.16) We will see that this vector is nonzero since, by hypothesis, the Jordan chain (v 1 , . . . , v 4 ) has length four. This is discussed below in §6. Solvability of the (Ω i ) T terms leads to the matrix term The terms containing δ i give The terms involving (ψ i ) XT are similar to those seen at fourth order, and generate The coefficient of the nonlinear term U U X simplifies to (5.20) When c g = 0 the vector function H(ζ, ζ) reduces to H(ζ, ζ) = D 2 k B(ζ, ζ) which is the form found in reduction of multiphase modulation to the KdV equation in [33,31].
Collecting these terms gives the vector form of the solvability condition for (5.15) This equation is interesting in itself, but it is not closed due to the presence of the δ XX term and the ψ XT term. However, the δ XX term is acted on by E ( c g ) and so this term vanishes when the equation is projected onto the kernel of E(c g ). Therefore split R 2 as and R N = span{ζ} ⊕ R N −1 when there are N phases. The projection of (5.21) onto the complement of Ker(E(c g )) still contains the δ XX term but this part carries over to higher order. With this splitting in mind, act on (5.21) with ζ T , and noting that the coefficient of the δ XX term now vanishes as E(c g ) is symmetric, (5.22) simplifies the vector equation to This equation is closed by first differentiating with respect to X, and applying conservation of waves and the ψ-U equation (5.12), Hence the final form of the two-way Boussinesq equation is Another way to write this is to use E(c g ), µ = 1 2 ζ T E (c g )ζ + ζ T E (c g )γ ; (5.26) emphasising that µ = 0 is the termination condition for the (ζ, γ) Jordan chain. Comparing ζ T H(ζ, ζ) with (2.24) shows that The emergent two-way Boussinesq equation is non-degenerate when µ, κ and K are nonzero. The coefficient µ is nonzero when the Jordan chain for E(c g ) in (4.5) terminates at two. The coefficient κ is assumed to be nonzero. If it is zero, then it is expected that re-modulation will lead to a cubic nonlinearity [14,34,31]. The coefficient of dispersion is nonzero if the Jordan chain on the left in (6.10) terminates at four. If K vanishes, then a longer Jordan chain will emerge. Re-modulation in this case is expected to lead to higher order dispersive terms emerging (e.g. sixth-order dispersion, as in [37,29]). The above result does not provide any information about convergence of the ansatz (5.4) as a Taylor series in ε. However, the asymptotic validity of this ansatz is confirmed by the above results; that is, the ansatz (5.4) satisfies the governing equations exactly up to O(ε 5 ), For generic multiphase WMT, a rigorous proof of validity has been given for CNLS [8], but a rigorous proof of validity in the case of coalescing characteristics is an open problem.
To summarize, the starting point is a PDE generated by a Lagrangian with a multiphase basic state. It is assumed that, at some parameter value, a pair of coalescing characteristics arises in the linearized Whitham equations. These coalescing characteristics generate several Jordan chains. A modulation ansatz of the form (5.4) then leads to a scalar two-way Boussinesq equation (5.24) with coefficients µ, κ, and K all determined from abstract properties of the averaged Lagrangian. The fundamental idea is that the original PDE is reduced to a simpler PDE that can be analyzed in some detail. Some of the solutions of this reduced two-way Boussinesq equation are anticipated in §7.

Coalescing characteristics and Jordan chains
It is clear that Jordan chains play an important part throughout the steps of the nonlinear modulation. In this section the properties of these Jordan chains are examined in more detail.
There are two key linear operators. The operator L, associated with the linearization of the Euler-Lagrange equation (2.3), generates a Jordan chain theory that starts with The operator E(c g ), the linearisation of the generic multiphase WMEs, generates another Jordan chain which can be discussed independently of the L−chains, but feeds into solvability of the L chains, and it starts with E(c g )ζ = 0 . (6. 3) The theory needed to extend these two Jordan chains is well established in the literature. The above L−chains are J−symplectic Jordan chains and this theory goes back to Williamson [45], and the theory of Jordan chains for quadratic Hermitian matrix pencils is developed in [16]. However, things get complicated when we realise that the linear operator L has both J−Jordan chains and M−Jordan chains. From (2.11) it follows that there exist M−Jordan chains of the form which follow from (2.11) with The J−chains (6.2) have length greater than two if either is solvable, and termination at two is associated with non solvability. These two chains can also be mixed, by taking the first elements to be linear combinations of ξ 1 and ξ 2 . Similarly, the M−chains have length greater than two if either is solvable, and termination at two is associated with non solvability. Again, these two chains may be mixed by taking the first elements to be linear combinations of ξ 1 and ξ 2 . Combining all the possibilities for both J−chains and M−chains, the most general extension of the Jordan chains is that there exists a vector Ξ satisfying (6.5) No theory exists for Jordan chains of this type. The closest approximation is the Jordan chain theory for multiparameter eigenvalue problems (e.g. Binding & Volkmer [3] and its citation trail), but that does not apply here either. We will be able to develop a satisfactory theory for multi-dimensional Jordan chains of this type to cover the cases needed in the modulation theory, but a complete and general theory for multi-dimensional Jordan chains of this type is outside the scope of this paper. A solution Ξ of (6.5) exists if this equation is solvable, and it is solvable if and only if the eight constants a j , b j , c j , d j for j = 1, 2, satisfy both or, using the identities in §2.1.2, In vector notation, this is Hence, if there exists values of these eight constants for which the equation (6.6) has a non-trivial solution, then Ξ is the next vector in the generalised Jordan chain. Interestingly, solvability of an equation with L is related to solvability of a reduced system on R 2 , which we will see then feeds into solvability of an equation with E(c g ). A general theory considering all possible Jordan chains emanating from the condition (6.6) is outside the scope of this paper. We will highlight special cases that appear in the nonlinear modulation theory. The case a = c = 0 (a pure J−chain) appears in the nonlinear modulation theory associated with zero characteristics [33,28,30], and the case b = d = 0 is mathematically equivalent, and generates a pure M−chain. Here two new cases which intertwine the J and M chains, and are required for the nonlinear modulation theory in this paper, will be highlighted.

A mixed Jordan chain with 4 ⊕ 2 structure
Taking a = c 2 ζ , b = c = cζ and d = ζ , (6.7) reduces the solvability condition (6.6) to That is; one case where (6.5) is solvable for Ξ is to take a, b, c, d in the form (6.7) with ζ ∈ Ker[E(c)]. In this case, the Jordan chain associated with L can continue when ∆(c) = 0 and ζ ∈ Ker[E(c)], the familiar condition (1.21) for the existence of a characteristic c. However, this construction does not imply that c = c g , that equivalence will follow from another Jordan chain and it is considered in §6.2.
In the case (6.7) with (6.8) the Jordan chain intertwines the symplectic J−chain and the symplectic M−chain. They can be combined to a new symplectic Jordan chain, based on the combined symplectic operator J + cM and ultimately leads to a new chain, which shows up in the nonlinear modulation theory.
Suppose first that c is arbitrary, and see that the condition c = c g will arise as a condition to extend the Jordan chain. With c arbitrary there is still a geometric eigenvector ζ satisfying E(c)ζ = 0. Express it in components, ζ = (ζ 1 , ζ 2 ), and re-number the generalized eigenvectors as follows, Substituting for v 2 using (6.9) and combining these two equations generates the equation (6.8). Indeed it was working backwards from (6.8) that suggested the definitions (6.9). Since L is symmetric and J + cM is skew-symmetric every Jordan chain has even length, assuring the existence of v 4 . It is assumed that the two chains (6.10) terminate at four and two respectively. Hence the systems Lv 7 = (J + cM)v 4 and Lv 8 = (J + cM)v 6 , (6.11) are not solvable. Non-solvability of the second condition in (6.11) is satisfied when the geometric multiplicity of c as an eigenvalue of E(c) is one; in the case of two-phase wavetrains this condition is Trace(E(c)) = 0. Non-solvability of the first equation in (6.11) arises in the nonlinear modulation theory to generate the coefficient of dispersion; that is, K = 0.

Another mixed Jordan chain defining c g
There is yet another Jordan chain, associated with L, that arises in the nonlinear modulation theory and the solvability condition for this chain defines c g . It is a special case of the solvability condition (6.6) obtained by taking Replacing c by c g anticipates the outcome of the solvability condition. Substitution of (6.12) into (6.6) and rearranging gives This equation is satisfied precisely when c = c g and then (ζ, γ) form a Jordan chain for E(c g ) of length two. The solvability condition ζ, E (c g )ζ = 0 defines c g as shown in (4.5) and (4.7). Suppose the solvability condition (6.13) and (6.14) is satisfied, then substitution back into (6.5) gives that It is this equation that arose in the modulation theory at fourth order (5.14) and working backwards we see that it is a special case of (6.5) and moreover solvability, with the expressions (6.12), is precisely the condition for the Jordan chain (6.14) of E(c g ). Further still, we can define another special case, which results in the criterion for the termination of this chain. This is achieved by setting Utilising this in (6.6) and simplifying results in the system E(c g )Υ + E (c g )γ + 1 2 E (c g )ζ = 0 , which is precisely (4.9). The assumption made here is that this chain is of length two, and so the right hand side of (4.9) does not lie in the range of L. Thus, by appealing to solvability one recovers the condition that , and therefore completing the connection between µ and the termination of this mixed Jordan chain. Within the modulation theory, this corresponds to the system being unsolvable for Γ, and what ultimately leads to the coefficient of the time derivative term in the emergent two-way Boussinesq.
We have only scratched the surface of the possible solvability conditions and attendant Jordan chains associated with (6.5). However, we have all the Jordan chains needed for the nonlinear modulation theory.

Properties of the two-way Boussinesq equation
Once the modulation equation is derived in a specific context, analysis of the solutions of the two-way Boussinesq equation (5.24) gives information about the nature of solutions in the nonlinear problem near coalescence.
The two-way Boussinesq equation is valid at c = c g . At least one parameter needs to be varied to obtain the coalescence. That parameter can be a one-parameter path through the four-dimensional frequency-wavenumber (ω, k) space, or it could be a perturbation of the frame speed c = c g + O(ε 2 ). Unfolding the singularity generates a term of the form νU XX in (5.24), regardless of the precise perturbation path (this can be shown by perturbing the linearised generic Whitham equations). Therefore the full modulation equation in the neighbourhood of the coalescence is where ν is an order one constant. When the coefficients are non-zero, the Boussinesq equation can be put into standard form. Scale the independent and dependent variables: τ = aT , ξ = bX, and U = ρu; then values of a, b, ρ can be chosen so that the two-way Boussinesq equation becomes with s 1 = sign(µν) and s 2 = sign (µK ) .
The sign s 1 determines whether the unfolding is into the elliptic region (s 2 = +1) or into the hyperbolic region (s 2 = −1, in which case all characteristics are hyperbolic). The sign s 2 indicates whether the resulting two-way Boussinesq equation is good (s 2 = +1) or bad (s 2 = −1). In the latter case, the initial value problem for the linearized system about u = 0 is ill posed, and small initial data with zero mean is therefore expected to saturate to form nonlinear structures. The ill-posedness in the case s 2 = −1 can be seen by considering the linearization of (7.2) about the trivial solution and introducing a normal mode solution of the form e i(kξ+ωτ ) . The dispersion relation associated with the normal mode is then There are four cases depending on the signs s 1 and s 2 , and they are shown in Figure 2. The figure plotsω 2 againstk 2 and soω 2 < 0 indicates linear instability of the trivial solution which in turn reflects linear instability of the basic travelling wave. When s 1 < 0 (the upper two cases in Figure 2) then either an unstable band emerges at finitek when s 2 = −1 or the Boussinesq equation is hyperbolic for all wavenumbers (s 2 = +1). When s 1 > 0 (lower two cases in Figure 2) then either a cutoff wave number emerges with re-stabilization at finitek (as in the lower right diagram with s 2 = +1), or instability is further enhanced for all wavenumbers (s 1 = +1 and s 2 = −1).
Integrating and taking the function of integration to be constant The constant of integration h is fixed by initial data or the value of u at infinity. For appropriate parameter values, this planar ODE has a family of periodic solutions and a homoclinic orbit which represent periodic travelling waves and a solitary travelling wave solution of (7.2). The implication of these solutions is that the transition from elliptic to hyperbolic of a periodic travelling wave of the original system generates a coherent structure in the transition, which is represented by the above solitary wave. However, there is much more complexity generated at the transition. Hirota [19] shows that there is a large family of N −soliton solutions to (7.2) as well. Further details especially in the case N = 2 are given in [19]. Blow-up can occur in the two-way Boussinesq equation even in the case of the good Boussinesq equation [39]. It is also generated by a Lagrangian, and has both a Hamiltonian and multisymplectic structure (e.g. §10 of [6] and [12]).

CNLS wavetrains with coalescing characteristics
To illustrate the theory it is applied to the modulation of two-phase wavetrains of a coupled nonlinear Schrödinger (CNLS) equation. This example serves two purposes: firstly, it shows that the coalescence of characteristics is quite common and appears even in the simplest of examples, and secondly, it shows that computing the coefficients in the emergent two-way Boussinesq equation is elementary once the properties of the basic state are found. The CNLS equation is a canonical example of a PDE generated by a Lagrangian with a toral symmetry, T 2 = S 1 × S 1 . Indeed any finite number of NLS equations can be coupled together to generate a toral symmetry T N for any natural number N , and they will have explicit N −phase wavetrains which are also relative equilibria. Here attention is restricted to two coupled NLS equations in the form where the coefficients α j , β ij , i, j = 1, 2, are given real constants, with β 21 = β 12 . The functions Ψ j (x, t) are complex-valued and i 2 = −1.
Coupled NLS equations appear in a wide range of applications. Two applications that motivated this work are the coupled NLS equations that appear in the theory of water waves (e.g. Roskes [35], Ablowitz & Horikis [2], Degasperis et al. [13]), and in models for Bose-Einstein condensates (e.g. Salman & Berloff [36], Kevrekidis & Frantzeskakis [21]). The PDE (8.1) is the Euler-Lagrange equation for with Ψ := (Ψ 1 , Ψ 2 ) and with the overline denoting complex conjugate. The toral symmetry follows from the fact that (e iθ 1 Ψ 1 , e iθ 2 Ψ 2 ) is a solution of (8.1), for any (θ 1 , θ 2 ) ∈ S 1 ×S 1 , when (Ψ 1 , Ψ 2 ) is a solution. The complex coordinates can be converted to real coordinates, generating a standard action of T 2 but will not be needed as the main calculations can be done in the complex setting.
Noether's theorem gives the conservation laws The basic state is just the usual family of plane waves, but interpreted here as a family of relative equilibria associated with the T 2 symmetry; it has the form, Substitution into the governing equations (8.1) generates the required relationship between the amplitudes, frequencies and wavenumbers, with β 21 := β 12 and β = β 11 β 22 − β 12 β 21 = 0.
The key wave action vectors A(ω, k) and B(ω, k), needed for analysis of the linearization, are obtained by substituting (8.5) into the components of the conservation law (8.3), and .
The linear operator E(c) defined in (1.18) is and and The characteristic polynomial is with Coalescing characteristics are obtained by solving ∆(c) = ∆ (c) = 0 for c. This problem is solved numerically in [11] by using graphical sign characteristic. The function ∆(c) is plotted versus c as parameters vary. That way roots and points where ∆ (c) = 0 can be read off the graph. It is inspired by the graphical Krein signature introduced by Kollar & Miller [22]. Results in [11] show that coalescing characteristics are plentiful in the Whitham modulation theory for CNLS. According to the theory in this paper at coalescing characteristics the following nonlinear modulation equation is generated In principle the quartic ∆(c) = 0 can be solved in closed form, but in practice this is lengthy and not illuminating, and numerical methods are more effective. For simplicity here, the case of standing waves (where k = 0) are considered, which restricts the parameter space significantly, and so calculations can be done explicitly. The strategy for calculating µ and κ is to construct the averaged Lagrangian and use the formulas (1.26) and (2.25).

Calculations for standing waves
Standing waves are defined as basic states of the form (8.4) but with k = 0. With this restriction the coefficients a 1 and a 3 are identically zero reducing the coefficients in the polynomial in (8.12) to There are four characteristics and they satisfy the biquadratic equation Coalescing characteristics occur precisely when the discriminant vanishes One way to interpret this equation is as a line in the positive quadrant of |Ψ 0 which includes the conditions β < 0 and α 1 α 2 < 0 for reality. At coalescence it follows from (8.14) that c 2 g = −α 1 β 11 |Ψ 0 1 | 2 − α 2 β 22 |Ψ 0 2 | 2 , which carries with it the requirement that α 1 β 11 |Ψ 0 1 | 2 + α 2 β 22 |Ψ 0 2 | 2 < 0, a condition that is effectively a generalisation of the defocussing classification for the one-component NLS. Now suppose parameters are such that (8.15) is satisfied, and proceed to compute the required coefficients in (8.13). The eigenvector and generalised eigenvector of E(c g ) are, Now use these eigenvectors and the Jacobians (8.8), (8.9) and (8.10) to generate the coefficients of the emergent Boussinesq equation. The first computed is the coefficient of the time derivative term, Next, one may use the variation of the Lagrangian to show that the coefficient of the nonlinear term is The coefficient of dispersion requires a Jordan chain analysis. This would require multisymplectification of CNLS and construction of the linear operator L. However, this CNLS has been multisymplectified in Ratliff [28], where reduction to KdV and 2-parameter Boussinesq were studied, and the Jordan chain theory is close to this case. With minor modification of that analysis, the desired dispersive coefficient is found to be .
Each of these coefficients has a common factor κ 0 , and so the two-way Boussinesq that emerges at the coalescence of characteristics simplifies to .
Withκ andK nonzero, one can proceed to analyze the solutions of this equation using results in the literature (e.g. [19,39]). A detailed analysis of (8.17) and its implications for coupled NLS is outside the scope of this paper, but the diversity of complexity due to coalescing characteristics is clear; for example, evaluation ofK along the lines (8.15) shows that (8.17) can be both positive (good Boussinesq) and negative (bad Boussinesq). with ·, · an inner product on C N . Guo & Lancaster [18] study quadratic eigenvalue problems in general and applying their definition to (9.1) gives the following.
Definition. The quadratic Hermitian matrix pencil (9.1) is hyperbolic if β 2 > αγ for all nonzero u ∈ C n . If this condition is satisfied then all the characteristics are real, and no coalescence can occur. It is expected that the absence of coalescence would be rare. The CNLS example shows coalescence to be quite common, already with N = 2. For arbitrary N , the parameter space (ω, k) has dimension 2N and so there is a high probability of coalescence. On the other hand, the above definition is a useful starting point in the analysis of multiphase WMEs. In the paper [18] they go on to give a number of sufficient conditions, and an algorithm for testing hyperbolicity and computing all the eigenvalues.
There is a known case where multiphase Whitham equations are hyperbolic. The paper of Willebrand [44] derives the multiphase WMEs and takes the limit N → ∞ and argues that they are hyperbolic in this limit. The argument proceeds by formally constructing explicit expressions for the leading order nonlinear corrections. Small divisors and divergence are expected, but only the leading order terms are studied. When N is small, "splitting of group velocity" is noted in the weakly nonlinear case, which is equivalent to what is called "coalescing characteristics" in this paper. The unfolding of this split group velocity may lead to instability. But Willebrand argues that the splitting disappears as N → ∞. In the context of this paper the limit N → ∞ would just replace the matrix pencil E(c) by an Hermitian operator pencil and so Willebrand's claim would be that E(c) in the case N → ∞ is hyperbolic. It is important to keep in mind that this argument is for multiphase modulation of weakly nonlinear Stokes waves only, but is an intriguing example nevertheless.
The above results clearly point to some interesting open problems in multiphase Whitham modulation theory: proving hyperbolicity, algorithms for computing characteristics and coalescence of characteristics, and studying the cases of large N and the limit N → ∞.

Concluding remarks
The theory of this paper has illustrated that coalescing characteristics create nonlinear dispersive dynamics, and it transpires that the resulting normal form is the two-way Boussinesq equation. Although we have confined the discussion to 1 + 1 dimensions, there is a natural generalisation to 2+1. A good starting point is the 2+1 theory for the nonlinear modulation of single-phase wave trains near coalescing characteristics [10]. However, the Jordan chain theory in (6.5) will literally take on a new dimension, bringing in the intertwinement of three symplectic Jordan chains. On the other hand, key features like the frame speed, scaling, and reduction will carry over with appropriate modification.
Whitham theory can also be formulated relative to any moving frame, and some frame speeds are more interesting than others (Ratliff [32]); indeed, it is found there that Whitham theory, when re-modulated relative to a characteristic frame speed, can generate dispersion, on a long enough time scale.
The results in the paper are universal, and are operational whenever a Lagrangian system has a suitable characteristic collision, which can be identified via the sign characteristic diagnostic used in [11] and in this paper. There are examples in the literature where multiphase Whitham modulation theory has been applied and coalescing characteristics observed, and so the application of the theory in this paper is relevant. Two examples are Stokes travelling waves coupled to meanflow (Whitham [42], Willebrand [44]) and modulation of viscous conduit periodic waves (Maiden & Hoefer [25]). Both of these examples have special features which require additional methodology. In the case of viscous conduit waves [25] the equations are not generated by a Lagrangian so the theory would have to be built on averaging of conservation laws. However at coalescing characteristics one expects a two-way Boussinesq equation to be generated or analogous equation with additional non-conservative terms. The case of modulation of Stokes waves in shallow water [42] involves the full water wave problem and so the class of PDEs (2.3) has to be modified to account for the vertical variation of water wave fields. However the full water wave problem has a multisymplectic structure (e.g. Chapter 14 of [4]) and so the theory should go through as in this paper, with appropriate modification. In the case of modulation of Stokes waves, the theory would potentially generate a new asymptotically valid two-way Boussinesq equation in shallow water hydrodynamics.
The focus of this paper has been on the case of two-phase wavetrains, but the results presented here can be extended quite naturally to problems involving arbitrarily many phases. All that changes is the dimensionality of the matrices and vectors involved, so long as the geometric multiplicity of the zero eigenvalue of E is one, and the other assumptions on the linear problem are satisfied. As noted above, the limit N → ∞ is intriguing.
On the other hand, a rather different problem arises when the kernel of E(c) has dimension greater than one. In this case, the secondary reduction to span{ζ} would be modified to span{ζ 1 , . . . , ζ k } where k ≤ N is the dimension of the kernel of E(c). Then k−additional coupled modulation equations are generated (one linked to each kernel direction).
Even in the case of two phases the parameter space is at least four dimensional, involving ω 1 , ω 2 , k 1 , k 2 , with further degrees of freedom emerging when system parameters are present. Hence higher order singularities are to be expected, e.g. more than two characteristics coalescing, or the coefficients µ, κ, and K passing through zero. A potential rescaling and re-modulation could then be implemented leading to (as yet unknown) modulation equations replacing the two-way Boussinesq equation.