## Abstract

The Kadomstev–Petviashvili (KP) equation is a well-known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper, it is shown that the KP equation is also the relevant modulation equation for bifurcation from *periodic travelling waves* when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a universal form, with the coefficients determined by the components of the conservation of wave action. The theory is derived for a general class of partial differential equations generated by a Lagrangian using phase modulation. The theory extends to any space dimension and time, but the emphasis in the paper is on the case of 3+1. Motivated by light bullets and quantum vortex dynamics, the theory is illustrated by showing how defocusing NLS in 3+1 bifurcates to KP in 3+1 at criticality. The generalization to *N*>3 is also discussed.

## 1. Introduction

The Kadomstev–Petviashvili (KP) equation in 3+1 can be scaled so that it takes the form
*N*+1 with *N*>3 one just adds additional second derivative terms for each new space dimension on the right-hand side. The case *N*=2 is the classical KP equation first derived in [1]. There has been a vast amount of work on 2+1 KP and a review can be found in Biondini & Pelinovsky [2]. The 3+1 KP has been much less studied. It first appeared in the paper of Kuznetsov & Turitsyn [3], where they study the transverse instability of 2+1 lump solitary waves in the 3+1 KP equation, showing that they are unstable. Further work, including further detail on the instability of lumps in 2+1→3+1, as well as direct numerical simulation, is reported in Senatorski & Infeld [4] and Infeld *et al.* [5] (see also Infeld & Rowlands [6]). A range of exact solutions of 3+1 KP have been discovered (e.g. Ma [7] and references therein).

The interest in this paper is not in solutions or structure of 3+1 KP. The contribution of this paper is threefold. We show how and why the KP equation (in any dimension) arises, without recourse to a dispersion relation. The key assumption is that the wave action flux (with the number of components dependent on dimension) has a critical point in wavenumber space. Secondly, it is shown that it is the relevant modulation equation for *periodic travelling waves* with critical wave action flux. Indeed, it would be quite complicated to construct the dispersion relation in general for the linearization about a family of periodic travelling waves, yet the approach based on criticality of the wave action flux is straightforward. Thirdly, we are able to predict the coefficients without recourse to any specific equation, they just follow from the structure of the Lagrangian, and the conservation of wave action. This latter aspect of the theory is reminiscent of Whitham modulation theory (e.g. ch. 14 of [8]), but here the modulation generates dispersion.

We assume that the partial differential equations of interest are generated by a Lagrangian
*Z*(*x*,*y*,*z*,*t*). The Lagrangian (1.2) is for the 3+1 case but has obvious extension to higher space dimension.

Suppose there exists a periodic travelling wave solution of the Euler–Lagrange equation
*θ*_{0} is an arbitrary phase shift, *ω* is the frequency and **k**:=(*k*,*m*,ℓ) is the wavenumber vector. We assume existence and smoothness of this family of periodic travelling waves.

The form of the emergent KP equation arises by a phase modulation argument. First, the frequency and wavenumber are made explicit in the basic state: replace (1.3) by *ϕ*,*q*,*r*,*s*,*Ω* are all functions of *X*,*Y*,*Z*,*T*,*ε*. Although the combination of scales in (1.4) looks strange, it is in fact naturally dictated by the *conservation of waves*, coupled with the scalings (1.11). Recall the classical way to define the local wavenumber and frequency for a given phase

The expression (1.4) is an ansatz. The strategy is just to substitute (1.4) into the Euler–Lagrange equation associated with (1.2), expand everything in powers of *ε* and equate terms of each order to zero. We find that the governing equations are satisfied exactly up to fifth order in *ε* if and only if *q* satisfies (1.10).

This strategy of ‘phase dynamics’ goes back to Whitham (e.g. ch. 14 of [8]) and the Whitham modulation theory, which was based on a Lagrangian. An inspiration for this work was the reduction theory of Doelman *et al.* [9] which suggested modulating parameters as well as the phase, but that theory involved reduction of reaction–diffusion equations. The theory came full circle in the work of Bridges [10,11] where modulating parameters and new scaling was included in the Lagrangian setting giving a new approach to modulation in the conservative setting. This theory led to a new universal form for the codimension one (only one assumption needed) emergence of the KdV equation. In [11], the KP-II equation in 2+1 is derived using phase modulation around steady solutions of the water-wave problem. An introduction to modulation in the conservative setting is given in [12].

It follows from the Whitham theory that the Lagrangian has a conservation law for wave action [8], ch. 11 and 14, which we write as
*A* is the wave action, and **B**:=(*B*,*C*,*D*) is the wave action flux vector. The (*A*,*B*,*C*,*D*) in roman are the components of the conservation law considered as functions of *Z*(*x*,*y*,*z*,*t*). In the subsequent theory, it is these components *evaluated on the basic state* that are important. Define the wave action evaluated on the basic state (1.3) as
*ε* (where *ε* is a small parameter to be defined). In this equation, *T*, *X*, *Y* and *Z* are slow time and space scales
*ε*^{2}. In (1.10), the dependent variable *q*(*X*,*Y*,*Z*,*T*,*ε*) arises as a modulation of the *x*-direction wavenumber *k*. The assumption *qq*_{X}, and the assumptions

The remarkable feature of the 3+1 KP in (1.10) is that the coefficients other than

The strategy of this paper—introduce an ansatz, substitute into the Euler–Lagrange equation, derive exact equations up to fifth order and show that the coefficients are determined by a conservation law—is similar to [10] and so we will be brief, highlighting those features that are new and different. Indeed, the first three terms in (1.10) are the same as in [10]. The two key new features are the form of the transverse dispersion, and the fact that the reduction in the *N*+1 case is codimension *N* (e.g. (1.9)). Although we will show that the codimension can be reduced when the system has a transverse reflection symmetry.

Our principal example is the reduction of the 3+1 defocusing nonlinear Schrödinger (NLS) equation to the 3+1 KP equation. The defocusing NLS in 3+1 has solitary wave solutions that are known as bullets due to their localized form in three space dimensions and they have attracted recent interest [13]. Although normally found in NLS with variable coefficients, the 3+1 KP has localized solutions that are similar to bullets [5], capturing a reduction of the three-dimensional localized solutions in defocusing NLS [14]. Another motivation for studying 3+1 NLS is quantum vortices (cf. Kerr [15] and references therein). In §8, we show how the theory in this paper gives immediately the coefficients in the 3+1 KP derived from 3+1 NLS. This reduction generalizes the reduction of NLS in 2+1 to KP-I (e.g. [16–18]).

An outline of the paper is as follows. The Lagrangian set-up, including structure of the Lagrangian, averaging, linearization and the conservation of wave action are introduced in §§2 and 3. Sections 4–6 give details of the modulation expansion and ordering of terms. When the system has a transverse reflection symmetry, the codimension of the emergence of KP is reduced by each such symmetry. The argument behind this is sketched in §7. The calculations giving rise to the reduction from 3+1 NLS to 3+1 KP are given in §8. In §9, the extension to any space dimension *N*>3 is outlined.

## 2. From Lagrangian to multisymplectic Hamiltonian

It is easier to proceed with the theory when the Lagrangian has structure. The strategy is to transform the Lagrangian density to a multisymplectic Hamiltonian density [19,20]. In this formulation, the conservation of wave action is given a geometric formulation [21] with a direct link to the equations.

The transformation from Lagrangian to multisymplectic Hamiltonian is effectively a multiple Legendre transform. Start with the *Lagrangian* formulation for some PDE
*U*(*x*,*y*,*z*,*t*) is in general vector valued. Legendre transform *V* =*δL*/*δU*_{t}, giving a *Hamiltonian formulation*
*W*=(*U*,*V*), and 〈⋅,⋅〉 an appropriate inner product, with **M** and *H* defined by Legendre transform. The density is still the same Lagrangian density with new coordinates. The advantage is that it has been split into two parts: a Hamiltonian function *H*(*W*_{x},*W*_{y},*W*_{z},*W*) which is scalar valued, and a part defined by a symplectic operator **M**, which for the purposes of this paper can be taken to be a constant skew-symmetric matrix.

Now continue to Legendre transform the Hamiltonian function in each space direction, resulting in a *multisymplectic Hamiltonian* formulation
*Z*(*x*,*y*,*z*,*t*), and 〈⋅,⋅〉 an appropriate inner product. The density is again the same Lagrangian density in terms of the new coordinates, but now it is split into *N*+2 parts, where *N* is the space dimension: a new Hamiltonian function *S*(*Z*) which does not contain any derivatives with respect to *t*,*x*,*y*,*z* and *N*+1 symplectic structures represented by the skew-symmetric matrices **M**,**J**,**K**,**P**. The principal advantage of the multisymplectic structure is that the symplectic structures appear both in the equations and in the conservation of wave action, giving an explicit connection for the modulation theory.

The above sequence of Legendre transforms is schematic, as in general non-degeneracy conditions are required, and each PDE has to be treated with care. An example of the above sequence of Legendre transforms is given in §8.

## 3. Euler–Lagrange equations and modulation

The starting point for the theory is the Euler–Lagrange equation associated with the Lagrangian (2.3)
*n*≥4. Here, **M**,**J**,**K**,**P** are constant skew symmetric matrices,
*k*,*m*,ℓ and frequency *ω*,
*θ*_{0}. Periodicity requires

The modulation ansatz is given in (1.4). The strategy is to substitute this modulation ansatz into the governing equations and equate like powers of *ε* to zero. First, preliminary results on the derivatives of the basic state and their connection with wave action conservation are established.

### (a) Averaging the Lagrangian and wave action

To get the components of the conservation law for wave action, average (2.3), evaluated on the family of travelling waves, over *θ*,
*ω*,*k*,*m*,ℓ, giving
*θ*,
**M**,**J**,**K**,**P** appear both in the Euler–Lagrange equations, (3.1) and in the components of the conservation law (3.3).

The derivatives in (1.9) are

We will also need the second *k* derivative of

### (b) Linearization about the periodic basic state

Define the linear operator
*θ* and *k*,
*ω*,*m*,ℓ which give

The first equation of (3.10) shows that **L**. It is natural to assume that the kernel is no larger. Hence assume
**L** with geometric eigenvector

For inhomogeneous equations that arise in the modulation theory and the Jordan chain theory, a solvability condition will be needed. With the assumption (3.12) and the symmetry of **L**, the solvability condition for the inhomogeneous equation **L***W*=*F* is

## 4. Details of the modulation expansion

The aim is to expand the modulation ansatz (1.4) in powers of *ε*, transform the derivatives using the chain rule, and then solve the equations at each order in *ε*. The small parameter *ε* is a measure of the distance in *k* space from criticality. Let *k*_{0} be a value of *k* satisfying *k*−*k*_{0}=*ε*^{2}*q* with *q* of order one. Taylor expanding the modulation of the basic state, we can write
*θ*,*k*,*m*,ℓ. Expand the remainder term *W* as well

The zeroth-order equation is just the equation for the basic state recovering (3.2). The first-order equation gives *q*=*ϕ*_{X}.

### (a) Third-order terms

At third order, terms proportional to *ϕ*^{3} and *qϕ* can be shown to vanish identically. For example, the terms proportional to *ϕ*^{3} are
*θ*. The *qϕ* terms vanish under a similar argument. This leaves
*r*=*ϕ*_{Y} and *s*=*ϕ*_{Z}. Then this system is considered solvable if
*k* in order to continue with the asymptotic analysis. This solvability condition confirms the first necessary condition in (1.9). The solution for *W*_{0} is then
*α*, and where *ξ*_{3} is defined through the relation

## 5. Interlude: Jordan chains

A Jordan chain of length *J*, {*ξ*_{1},…,*ξ*_{J}}, for a zero eigenvalue in the symplectic setting is defined by
**J** is invertible then this chain is a classical Jordan chain. However, in this case, **J** may not be invertible. Hence we include the assumption

Here we are interested in the Jordan chain associated with the geometric eigenvector **L***ξ*_{4}=**J***ξ*_{3} and this equation is solvable if and only if
**J**. Hence the Jordan chain has length at least four. There is no fifth element if
*x*-direction.

## 6. Terms of order four and five in the expansion

At fourth-order, the equation simplifies to
*q*_{Y}=*r*_{X} and *q*_{Z}=*s*_{X}, from the conservation of waves have been used.

Note that the first term vanishes if *Ω*=*ϕ*_{T}, a similar enforcement to the previous orders. The term prefactored by *q*_{X}*ϕ* can be shown to be the result of **L**(*ξ*_{3})_{θ} since if we differentiate its defining equation (4.4) with respect to *θ*,
*q*_{Y} term? Checking solvability:
*k* to continue. Similarly, to solve for the *q*_{Z} term, we consider its inner product:
*k* extremality in *W*_{1} at this order is

### (a) Fifth-order terms

After a few simplifications at this final order, we have the equation
*r*_{Y} coefficient is also simple to evaluate and gives the result
*p*_{Z} term. Via calculation:
*X* derivative of this equation gives (1.10) and thus the 3+1 KP equation governs the dynamics of the perturbation at fifth order.

The *qq*_{X} bracket in the above appears to be the *k* derivative of the *ξ*_{3} equation,
*qq*_{X}, vanishes identically. However, the *k* derivative for this vector does not necessarily exist: *ξ*_{3} only exists for specific values of the wavenumber, defined by *ξ*_{3} is not necessarily differentiable and so the coefficient of *qq*_{X} is, generically, non-zero.

### (b) Reduction to the 2+1 case

The most widely studied case of the KP equation is the 2+1 case [2]. The theory here reduces immediately to that case by restricting the original PDE to have coordinates (*x*,*y*,*t*) only. The 3+1 KP reduces to this case by neglecting *Z*-dependence giving

One of the most well-known contexts for the appearance of the 2+1 KP equation as a model equation is in the theory of water waves. A special case of the modulation approach was used in Bridges [11] to give a new derivation of the KP-II equation in shallow water, and showed the connection between the coefficients and the properties of classical uniform flows. The theory of this paper suggests that the KP equation may also appear as a modulation equation in water waves in the perturbation about non-trivial periodic travelling waves.

## 7. Implications of a transverse reflection symmetry

One of the curiosities of the emergence of the KP equation is that it is codimension *N* where *N*−1 is the number of transverse space directions (meaning that *N* conditions (1.9) are necessary for emergence). On the other hand, the KdV equation is codimension 1, requiring only the condition

This contradiction is rectified by noting that when the governing equations have a transverse reflection symmetry in the *y*-direction then the condition *m*=0. Similarly when there is a transverse reflection symmetry in the *z*-direction then the condition *y*-direction reflection symmetry. A similar argument works in any transverse direction.

The implication of a reflection symmetry for the solution set is that *Z*(*x*,−*y*,*z*,*t*) is a solution whenever *Z*(*x*,*y*,*z*,*t*) is a solution. This reflection symmetry will also arise in some form in the functions *y*-reflection is the following property:
*m* and *m*.

A system in multisymplectic form (3.1), is *transverse reversible* in the *y*-direction if there exists a *reversor* **R** acting on **R** is a *reversor* if it is an involution and an isometry.

Act on (3.1) with **R**,
**R***Z*(*x*,−*y*,*z*,*t*) is a solution of (3.1) whenever *Z*(*x*,*y*,*z*,*t*) is a solution.

We will verify the third of (7.1) as it is the most important with the verification of the others following a similar argument. Start with the definition
**R** and use (7.2) to establish that
**R****K**=−**K****R** and **R** is an isometry. This completes the verification of the third identity in (7.1), with the others verified in a similar manner. The fact that *m* gives immediately
*x*-direction. The example of NLS in 3+1 in the next section has a reflection symmetry in all transverse directions.

## 8. Example: 3+1 nonlinear Schrödinger equation

Consider the defocusing NLS equation in 3+1 in standard form
*ψ*(*x*,*y*,*z*,*t*). This system is a basis for the discussion of quantum vortices [15]. Separate the equation into real and imaginary parts by letting *ψ*=*a*_{1}+i*a*_{2}, giving
**a**=(*a*_{1},*a*_{2})^{T}. This then allows us to write the Lagrangian for this system in the form of (3.1) with

The associated conservation law of the form (1.7) can be deduced from (8.1) with
*ψ** denotes the complex conjugate of *ψ* and ℑ denotes the imaginary part of the expression. This conservation law can be verified by direct calculation using (8.1). The symbols (*A*,*B*,*C*,*D*) represent the components of the conservation law as functions of *ψ*. For the modulation theory, it is these components evaluated on the basic state that are important.

Consider a basic state of the form
**a**=(*a*_{1},*a*_{2}) and *a*_{1}+`i`*a*_{2}=*Ψ*_{0} ^{θ}. This state is an exact periodic travelling wave solution and substitution into the governing equation gives
*m* and *m*. Similarly, *y*- and *z*-directions, respectively. These symmetries follow from the theory in §7, or simply by noting that the *y*- and *z*-derivatives in 3+1 NLS (8.1) are even.

Seeking *k* extremality in *k*=0, the second two conditions give *m*=ℓ=0. The first condition then reduces to

All that remains is to compute the *x*-direction dispersion coefficient

Express the Jordan chain elements in the form
*θ*)*R*_{θ}=*σ**R*_{θ} and the latter expression is commutative. The sequence that produces the elements of the Jordan chain can be found to be
**a**_{0}=**b**_{0}=**c**_{0}=**d**_{0}=**0**, then we find that
*y*- and *z*-directions. By scaling *q*,*X*,*Y*,*Z*,*T* appropriately, the emergent KP equation (8.12) can be put into canonical form
*et al.* [5] and localized solitary waves in three space dimensions have been shown to form from perturbed exact two-dimensional lump solitons.

## 9. Emergence of Kadomstev–Petviashvili in *N*>3 space dimensions

The emergence of KP follows from two key structural properties: the multisymplectic form of the Euler–Lagrange equation (3.1), and the conservation of wave action in geometric form (3.3). Of secondary importance is the scaling and the necessary conditions (1.9). All of these requirements generalize to arbitrary space dimension. Although applications in space dimension *N*>3 are not obvious, it is straightforward to sketch the argument leading to KP in *N*+1.

Consider the following generalization of (3.1):
**x**=(*x*,*x*_{1},…,*x*_{N−1}) and **M**,**J**,**K**_{n}, *n*=1,…,*N*−1, are skew-symmetric matrices. A periodic travelling wave is of the form
**k**=(*k*,*k*_{1}⋯*k*_{N−1}), frequency *ω* and phase shift *θ*_{0}. Substitution of this ansatz into (9.1) results in the ODE

The components of the conservation law for wave action have the natural generalization
*i*=1,…,*N*−1.

Generalizing the modulation ansatz (1.4), introducing slow space scales of order *ε*^{2} in all the transverse directions, substituting into the Euler–Lagrange equation and computing terms up to fifth order in *ε*, leads to the following necessary conditions:

## Data Accessibility

This paper is comprised of theory only and contains no data.

## Authors' Contributions

The authors contributed equally. The germ of the idea came from the second author. The first author undertook the analysis and example, both checked by the second. Both authors contributed to the writing.

## Competing Interests

We declare we have no competing interests.

## Funding

The research reported in this paper is partially supported by the EPSRC under grant EP/K008188/1, as well as a PhD studentship funded by EPSRC.

- Received February 26, 2015.
- Accepted April 30, 2015.

© 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.