## Abstract

Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.

## 1. Introduction

There is a multitude of physical examples where multiphase wavetrains arise naturally. In general, they are quasi-periodic solutions and so may have small divisor problems in establishing existence. However, in the case where there is some underlying symmetry, multiphase wavetrains can be characterized as relative equilibria and as such exist robustly in multiparameter families. For a vector-valued field, **u**(*x*,*t*), satisfying the Euler–Lagrange equation associated with a Lagrangian functional, an example of a multiphase wavetrain is a solution of the form
*k*_{1},*k*_{2}), (*ω*_{1},*ω*_{2}) are the, in general distinct, wavenumbers and frequencies.

In this paper, we are interested in the modulation of multiphase wavetrains in conservative systems. This problem was first studied by Ablowitz & Benney [1], using Whitham modulation theory. They derived the conservation of wave action for scalar fields with two phases in detail, and showed how the theory generalized to *N*-phases. Examples in Ablowitz [2] show that, in general, one should expect small divisors, but weakly nonlinear solutions could still be obtained. However, for integrable systems, multiphase averaging and the Whitham equations are robust and rigorous, and a general theory can be obtained [3]. On the other hand, if the system is not integrable, but there is an *N*-fold symmetry, then again a theory for conservation of wave action can be developed without small divisors and smoothly varying *N*-phase wavetrains [4]. In essence, the conservation of wave action is replaced by the conservation law generated by the symmetry. There is now a vast literature on multiphase averaging and dispersionless Whitham theory, but we are not going to pursue this direction further as the interest here is in how dispersion can be generated in the modulation of multiphase wavetrains.

The strategy, in this paper, for generating dispersion in modulation equations deduced from the multiphase conservation of wave action is motivated by the theory for the single-phase case studied in [5–7], but has subtle new features. The starting point is a general class of partial differential equations (PDEs) generated by a Lagrangian
*V* (*x*,*t*) is a vector-valued function of (*x*,*t*) on the rectangle [*x*_{1},*x*_{2}]×[*t*_{1},*t*_{2}]. It is advantageous to first transform the Lagrangian density to multisymplectic form
*x*,*t*) and *n* is assumed to be even. The structure operators **M** and **J** are constant skew-symmetric *n*×*n* matrices and

Suppose there exists a two-phase solution
*A*,*B*) are the components of the wave action conservation law when considered as functions of (*x*,*t*) and (**A**,**B**) when they are considered as functions of (**k**,** ω**), with components

**k**↦

**B**(

**k**,

**) fails to be bijective, in particular it is assumed that**

*ω***k**

_{0}(

**), and the zero eigenvalue is assumed to be simple with eigenvector**

*ω***k**

_{0}is dropped, and it is clear in context whether

**k**

_{0}or

**k**is intended. For each fixed

**, the condition (1.8) defines a curve in**

*ω***k**space which is called the criticality curve, or, as

**varies, the criticality surface. The matrix conditions (1.8)–(1.9) for emergence of dispersion generalize the scalar condition in the single-phase case. The condition (1.8) is a mathematical condition, but in §3f the connection with physical criticality is discussed.**

*ω*Given the basic state the strategy is to modulate using an ansatz, motivated by the one-phase case in [5–7], but generalized to two phases
*ε*, and solved order by order. The main result of this paper is that the *ε*^{5} equation is solvable if and only if the following equation is satisfied:
**K** and *U* is obtained by projection
** α** is unknown at this stage. Second, the coefficients of the first, second and fourth terms are all expressed purely in terms of derivatives of the components of the conservation law and the eigenvector

**, with the latter determined by (1.9).**

*ζ*On the other hand, D_{k}**B** is singular by assumption (1.8) and symmetric, so if the inner product of ** ζ** with equation (1.12) is taken, the last term drops out, and a scalar KdV equation emerges. Projection of (1.12) onto the complement of the kernel of D

_{k}

**B**provides an equation determining

**, which would be used at the next order.**

*α*Although this ansatz (1.10) looks like a straightforward generalization of the ansatz in the single-phase case, there are subtle non-trivial differences in the multiphase theory. The Jordan chain theory here (developed in §3b) intertwines the theory in the single phase case with the new eigenvector *ζ* in (1.9). It is not *a priori* clear how two solvability conditions generate one equation, and here it is the projection and the essential role of ** α** that are important (there is no analogue in the single-phase theory).

In summary, the theory starts with a conservative PDE, generated by a Lagrangian in canonical form (1.1)–(1.2), with a four-parameter family of two-phase wavetrains (1.4). When this four-parameter family has a simple degeneracy (1.8), the modulation ansatz (1.10) satisfies the Euler–Lagrange equation (1.3) up to fifth order in *ε* when (*U*,** α**) satisfy (1.12). Projection of (1.12) in the direction of the kernel of D

_{k}

**B**then generates a scalar KdV equation

There are many examples of conservative PDEs with multiphase wavetrains. In this paper, we include two representative examples. The first is two-layer shallow-water hydrodynamics with a free surface, and the second example is a coupled nonlinear Schrödinger (NLS) equation. In both cases, there is a natural two-parameter symmetry group, the first is associated with conservation of mass in each layer, and the coupled NLS has a natural toral symmetry. A multiphase wavetrain can be explicitly constructed in both cases and parameter values identified where the determinant condition (1.8) is satisfied. Potential generalizations are discussed in the concluding remarks section.

## 2. Governing equations and wavetrain properties

The governing equation is (1.3) with **M**^{T}=−**M**, **J**^{T}=−**J**, and, for simplicity, **J** is assumed to be invertible. Non-invertible **J** just requires an additional assumption on the kernel of **J**.

Suppose there exists a two-phase solution of the form (1.4). Substitution into (1.3) gives the following governing equation for the two-phase wavetrain:
** θ**,

**k**and

**. A class of systems for which this is true in general is symmetric systems when the multiphase wavetrains are relative equilibria associated with the symmetry, and both the examples in this paper have this property.**

*ω*Linearizing (1.3) about the two-phase solution leads to the linear operator

In the modulation analysis, equations for the derivatives of *a*
*b*
*c*The first of these equations highlights that the zero eigenvalue of **L** is not simple, and so we make the assumption that
*b*) and (2.4*c*)) suggest that there are two different Jordan chains, and these two chains will feature prominently in the modulation analysis. We also require the second derivatives with respect to wavenumber, and they can be summarized as
*a*and
*b*

The assumption (2.5) along with the formal self-adjointness of **L** give the solvability condition for

### (a) Conservation laws

When the Lagrangian is in canonical form (1.2), the components of the conservation law evaluated on the two-phase wavetrain, whether the conservation laws are deduced from symmetry or they are components of wave action conservation, have a geometrical form and can be readily deduced. The canonical Lagrangian (1.2), evaluated on the two-phase wave train and averaged, is
*a*
*b*
*c*Note that

## 3. Multiphase modulation leading to the Korteweg–de Vries equation

With the assumption (1.8), the proposed modulation ansatz is (1.10), with
*k*_{i},*ω*_{i} to *θ*_{i}. In particular, the above definitions impose that **q**_{T}=*Ω*_{X}, which is vector-valued conservation of waves. The strategy is to substitute the ansatz into the Euler–Lagrange equation (1.3), expand everything in Taylor series in *ε* and solve the equations at each order in *ε*. The Taylor expansion of *W* is

The *ε*^{0} equation just returns the equation for the basic state (2.1). The *ε*^{1} equation is
*a*), and so this is trivially satisfied, for any *ϕ*_{1} and *ϕ*_{2}, by the basic state. The *ε*^{2} terms are
*b*), we have
*q*_{1}=(*ϕ*_{1})_{X} and *q*_{2}=(*ϕ*_{2})_{X}.

### (a) Third order

At third order, the equations become more interesting. We find that
*b*),
*q*_{1})_{X}=(*q*_{2})_{X}=0), the requirement to proceed to the next order is that the 2×2 matrix D_{k}**B** must be singular. In the case where det[D_{k}**B**]≠0, and so **q**_{X}=0 leads to *W*_{3}=0 (mod the kernel of D_{k}**B**), the theory fails and dispersion does not emerge. When det[D_{k}**B**]≠0, the relevant modulation equation is dispersionless multiphase Whitham equations, albeit with a time scale *T*=*εt*.

It is at this point in the analysis where the assumption (1.8) becomes important. With the assumption that the zero eigenvalue of D_{k}**B** is simple with eigenvector (1.9), it follows from (3.3) that we must have
*U*(*X*,*T*). In principle, *U*(*X*,*T*) also depends on *ε*, but here only the leading-order term is needed so *U*(*X*,*T*)≡*U*(*X*,*T*,*ε*)|_{ε=0}. Integrating (3.4),
*V* (*T*) can be neglected as it does not affect the leading-order KdV equation that arises at fifth order.

The equation for *W*_{3} in (3.2) can now be cast into the form
** α**=(

*α*

_{1},

*α*

_{2}) is unknown at this point, and

*ξ*

_{5}is the solution of

**∈Ker[D**

*ζ*_{k}

**B**]. The solution for

*ξ*

_{5}coalesces the two Jordan chains in (2.4

*c*), leading to a 2⊕4 Jordan block structure. We digress here to identify further properties of the symplectic Jordan chain arising in this case.

### (b) Jordan chain theory interlude—multiple blocks

Jordan chain theory plays a key role in the modulation theory for the emergence of the KdV equation for the one-phase case in [5–7]. Here, a non-trivial generalization of that theory is required. There are two Jordan chains, but they intertwine. With the assumption (2.5), the zero eigenvalue of **L** has geometric multiplicity two. Hence, there are two Jordan blocks
*ξ*_{5} and, because all symplectic Jordan chains have an even number of elements, the next element is *ξ*_{6} satisfying

The left-hand chain terminates at two, due to the fact that zero is a simple eigenvalue of D_{k}**B**. It is assumed that the right-hand chain terminates at four. Define
**L***ξ*_{7}=**J***ξ*_{6} ensures that

### (c) Fourth order

After simplification, absorbing the relevant terms into the linear operator, and replacing **q** with *U*** ζ**, we find that

*ϕ*

_{1},

*ϕ*

_{2}cancel the

*Ω*

_{1},

*Ω*

_{2}terms by (2.4

*c*). Hence, using the top of the Jordan chain,

*W*

_{4}is

*γ*

_{1}(

*X*,

*T*) and

*γ*

_{2}(

*X*,

*T*) are arbitrary at this stage.

### (d) Fifth order

After simplification, the *ε*^{5} terms are
**L** as in (3.12). However, the explicit expression for

The key terms on the right-hand side are proportional to *U*_{T}, *U*_{XXX}, *UU*_{X} and *α*_{XX}, so the form of (1.12) is emerging. Application of the two conditions for solvability (2.7) produces two equations. The solvability conditions are applied term by term. The coefficients of the *U*_{T} term involve
**e** by
*U*_{T} is
*U*_{XXX}, use (3.10) and define *α*_{XX} terms is similar to the construction in §3*a* and solvability gives a term [D_{k}**B**]*α*_{XX}. By combining terms, and multiplying by −1, solvability of (3.15) gives
**f**=( *f*_{1},*f*_{2}), solvability of (3.15) gives, for *j*=1,2, that
**B**(**k**,** ω**), use (2.8

*c*) to obtain

*a*) with respect to

*k*

_{j},

_{k}

**B**then gives the scalar KdV equation (1.13).

In the one-phase case, the modulation equation arising at fifth order is precisely the KdV equation. Here, the intermediate equation (3.16) arises and it is indeterminate, because *α*_{XX} is unknown at this order. Hence, it is not a coupled KdV equation. It is at best an inhomogeneous coupled KdV equation with an indeterminate inhomogeneity. It is here that the eigenvector ** ζ** associated with the zero eigenvalue of D

_{k}

**B**plays a central role. Projection of (3.16) in the direction of

**then delivers a fully determined scalar KdV equation. Projection of (3.16) in the direction of the complement of the kernel of D**

*ζ*_{k}

**B**gives a defining equation for

*α*_{XX}, which is not needed until the next order.

The above analysis also gives an indication of how a true fully determined coupled KdV equation can arise: when D_{k}**B** has a double-zero eigenvalue with two independent eigenvectors, with the system once again emerging when one projects in the direction of the kernel.

### (e) Modulation of steady multiphase wavetrains

The reduction to KdV is essentially the same when the multiphase wavetrain is steady
**k** and so can be computed from the steady solution. This simplification is useful in examples.

The justification of this claim is as follows. The (*ϕ*_{i})_{T} terms in (3.11) now are no longer cancelled with the *Ω*_{i} terms, as these are not present, and so the additional solvability requirement is
**M** instead of **J**. Therefore, there exists *κ*_{i} such that
**e** is now given by

### (f) From mathematical criticality to physical criticality

The emergence of the KdV equation owing to degeneracy of the Jacobian D_{k}**B** is a mathematical result. The result is based on the abstract properties of a Lagrangian, and is deduced without any recourse to physical input. One interpretation of the importance of the mapping **k**→**B**(**k**,** ω**) can be seen by characterizing the Euler–Lagrange equation (2.1) as a constrained variational principle: multiphase wavetrains are critical points of the Lagrangian taking the wave action and wave action flux as constraints, then frequency and wavenumber are Lagrange multipliers. The values of

**k**,

**in the resulting solution**

*ω***k**,

**)↦(**

*ω***A**,

**B**) need to be non-degenerate in order to solve for (

**k**,

**) in terms of the values of the constraint sets. Breakdown of this mapping, det[D**

*ω*_{k}

**B**]=0, we call (mathematical) criticality, although we generally use the term ‘criticality’ without qualification.

All the steps in the theory follow from mathematical considerations. Hence, it is all the more remarkable that it is an excellent model for capturing criticality in physical systems. The connection is easy to see in simple models such as one-layer shallow-water flow where criticality corresponds to Froude number unity. The Froude number unity condition can than be re-characterized as the appearance of a zero eigenvalue which, in turn, can be related to mathematical criticality (see §8 of [5]). In the fluid mechanics literature, criticality, in more complicated systems, is often associated with the appearance of zero eigenvalues (see [9] for an extensive discussion of this viewpoint). It is the appearance of zero eigenvalues that connects the two theories: zero eigenvalues are an essential part of both mathematical and physical criticality.

While the connection can be made fairly precise in fluid mechanics, in other systems, such as the coupled NLS equations arising as a model for Bose–Einstein condensates and nonlinear optics, the connection is not as clear because of the absence of a concept of physical criticality. Nevertheless, the mathematical theory applies to give results which then have physical implications. Below, two examples are given. The first from fluid mechanics shows how the theory of this paper gives a new mechanism and simplified derivation of the emergence of KdV in two-layer shallow-water hydrodynamics. The second example shows how a single KdV equation can emerge at criticality for coupled NLS equations, showing how the mathematical theory can evoke new results for the physical system, without the need to understand the physical mechanism behind the criticality condition for NLS.

## 4. Example 1: two-layer shallow-water flow

A natural application of the multiphase theory presented here is the case of shallow-water hydrodynamics with two layers of differing density bounded above by a free surface. The conservation laws in this case are mass conservation in each layer, and they are associated with the potential symmetry. The basic multiphase wavetrain in this case is just uniform flow (constant depth and velocity in each layer).

The governing equations for this system are
*a*
*b*
*c*
*d*with
*ρ*_{1}, *η*, *u*_{1} are the density, depth and horizontal velocity in the lower layer, and *ρ*_{2}, *χ*, *u*_{2} are the density, depth, and horizontal velocity in the upper layer. In the dispersion coefficients, *η*_{0} and *χ*_{0} are quiescent depths in the two layers. The dispersionless version of these equations is derived in Baines [10], and the dispersive terms are derived in Donaldson [11] (see also [12]).

The first two equations (4.1*a*) and (4.1*b*) are the system’s conservation laws and the symmetry associated with them is a constant shift of the velocity potentials which are defined by *u*_{1}=(*ψ*_{1})_{x} and *u*_{2}=(*ψ*_{2})_{x}. Steady multiphase wavetrains associated with this symmetry take the form
*k*_{1} and *k*_{2} are constant velocities in each layer. Substitution into the governing equations requires that *k*_{1} and *k*_{2} satisfy
*R*_{1},*R*_{2} are constants of integration (Bernoulli constants in each layer). These two equations are used to express *η*_{0} and *χ*_{0} in terms of *k*_{1} and *k*_{2} in the conservation laws.

### (a) Conservation laws and criticality

The first two equations of this system (4.1*a*) and (4.1*b*) form the conservation laws for the system. Therefore, we have
*r*=*ρ*_{2}/*ρ*_{1}, which upon expansion is
_{k}**B**] in [14] was linked to the stability, with det[D_{k}**B**]<0 being necessary (but not sufficient) for flow instability. This loss of stability was shown to lead to the presence of hydraulic jumps within the flow. Assume that the trace of D_{k}**B** is non-zero, and so the zero eigenvalue is simple with eigenvector

### (b) Emergence of Korteweg–de Vries at criticality

The relevant coefficient matrices for the vector KdV equation (1.12) are
**K** is a coefficient of a linear term, it can be calculated using the dispersion relation (associated with the linearization about the basic state) or using the Jordan chain. The details are omitted and we just state the result,

## 5. Example 2: NLS × NLS → KdV reduction

Another example where a two-phase wavetrain arises naturally is the coupled NLS equation. To illustrate, we use a coupled NLS equation which appears in the theory of water waves [16] and in models for Bose–Einstein condensates [17]
*a*and
*b*for complex-valued functions *Ψ*_{1}(*x*,*t*) and *Ψ*_{2}(*x*,*t*) and real constants *α*_{i}, *β*_{ij}, with *β*_{21}=*β*_{12}. For convenience, define *β*=*β*_{11}*β*_{22}−*β*_{12}*β*_{21}≠0. There is a natural toral symmetry in that (*e*^{iθ1}*Ψ*_{1},*e*^{iθ2}*Ψ*_{2}) is a solution for any (*θ*_{1},*θ*_{2})∈*S*^{1}×*S*^{1} whenever (*Ψ*_{1},*Ψ*_{2}) is a solution. A two-phase wavetrain associated with this symmetry is
**k** and the amplitudes to be related by

### (a) Conservation laws and criticality

The components of the conservation law associated with the toral symmetry are
*A*,*B*) on the basic state
**B**(**k**)
_{k}**B** is non-zero and choose the eigenvector as
_{k}**B**]<0 has been shown to relate to the stability of these wavetrains. Examples of this instability criterion have been derived for specific systems by Roskes [16] and Law *et al.* [19]. A general result for conservative systems in terms of the conservation of wave action was derived in [4]. The condition det[D_{k}**B**]=0 is therefore of interest both as a necessary condition for emergence of the KdV equation and as an instability boundary.

### (b) Emergence of Korteweg–de Vries at criticality

The relevant matrices needed for the emergent vector KdV equation (1.12) are
*ζ*^{T}**K** can be calculated, using the dispersion relation from the linearization or by constructing explicitly the Jordan chain. The details are lengthy and will be given elsewhere. The result is

## 6. Concluding remarks

This paper has shown that systems generated by a Lagrangian that have two conservation laws and possess two-phase wavetrain solutions can lead to the emergence of a scalar KdV equation, when the mapping from wavenumber space into the flux vector is singular. It appears that the theory will generalize to *m*-dimensional Lie groups with *m*>1 when the group is Abelian, with the condition that D_{k}**B** have a simple zero eigenvalue, regardless of the dimension *m*. If the zero eigenvalue is not simple, then potentially a true vector-valued KdV equation is expected to emerge, but emergence of the coupled KdV is higher codimension, that is, it requires additional parameters.

Analysis of systems with a one-parameter Lie group has shown that, with additional conditions, other equations are generated by the modulation, e.g. the two-way Boussinesq equation [7] and the KP equation [12] in the 2+1 case. That theory should also generalize to the multiphase case. In the case of the Boussinesq equation, we would expect this to occur when the time term in the scalar KdV equation vanishes, which implies that the system

The analysis presented here appears to generalize quite naturally to multiphase wavetrains with any finite number of phases (with attendant symmetries and conservation laws). In the case of a simple zero eigenvalue, the results here are almost immediately applicable with some small alterations to accommodate the additional phases. The extra phases also more readily allow for situations where the geometric multiplicity increases and so the coupled KdV emerges as discussed.

There is a dual version of the theory by switching space and time. Then, it is degeneracy of the mapping ** ω**↦

**A**(

**k**,

**) that drives the theory, resulting in a scalar KdV equation with space and time reversed.**

*ω*## Data accessibility

Although there are no numerical data used in this paper, details of relevant data associated with the publishing and accessibility can be found on the University of Surrey publications repository at http://epubs.surrey.ac.uk.

## Authors' contributions

The derivation of the KdV, including a key breakthrough, and the introduction and analysis of the examples were contributed by D.J.R. The germination of the idea, and the development of the Jordan chain theory, were contributed by T.J.B. Both authors contributed to the writing of the paper.

## Competing interests

The authors have no competing interests.

## Funding

D.J.R. is fully supported by a EPSRC PhD studentship grant no. EP/L505092/1. T.J.B. is partially supported by EPSRC grant no. EP/K008188/1.

- Received June 7, 2016.
- Accepted November 7, 2016.

- © 2015 The Authors.

Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.