Whitham modulation equations, coalescing characteristics, and dispersive Boussinesq dynamics

Physica D 333 (2016) 107–116

Contents lists available at ScienceDirect

Physica D journal homepage: www.elsevier.com/locate/physd

Whitham modulation equations, coalescing characteristics, and dispersive Boussinesq dynamics Daniel J. Ratliff ∗ , Thomas J. Bridges Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK

highlights • • • • •

A theory for Whitham modulation with degenerate characteristics is developed. A novel method to obtain a universal two-way Boussinesq equation is presented. A new emergence of this Boussinesq equation from a complex Klein–Gordon model. The method provides insight into the Kelvin–Helmholtz instability. The method implies two-way Boussinesq model for water waves is invalid.

article

info

Article history: Received 19 June 2015 Received in revised form 26 October 2015 Accepted 11 January 2016 Available online 21 January 2016 Keywords: Nonlinear waves Modulation Lagrangian systems

abstract Whitham modulation theory with degeneracy in wave action is considered. The case where all components of the wave action conservation law, when evaluated on a family of periodic travelling waves, have vanishing derivative with respect to wavenumber is considered. It is shown that Whitham modulation equations morph, on a slower time scale, into the two way Boussinesq equation. Both the 1 + 1 and 2 + 1 cases are considered. The resulting Boussinesq equation arises in a universal form, in that the coefficients are determined from the abstract properties of the Lagrangian and do not depend on particular equations. One curious by-product of the analysis is that the theory can be used to confirm that the two-way Boussinesq equation is not a valid model in shallow water hydrodynamics. Modulation of nonlinear travelling waves of the complex Klein–Gordon equation is used to illustrate the theory. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The Whitham modulation equations are a hyperbolic or elliptic first order quasilinear system of partial differential equations that can arise from the modulation of a family of periodic wavetrains [1,2]. In the simplest 1 + 1 setting, with just conservation of wave action and conservation of waves, the Whitham modulation equations (WMEs) are a pair of partial differential equations (PDEs) with two characteristics. In almost all theory and application of Whitham modulation theory these two characteristics are assumed to be distinct. In [3,4] the case where one of the characteristics is zero was considered, and it was found that a rescaling and nonlinear unfolding generates dispersion in the WMEs and the conservation of wave action morphs into the Korteweg–de-Vries (KdV) equation. In this paper, the case of a double characteristic at zero is considered. With a rescaling, we find that the WMEs in

∗

Corresponding author. Tel.: +44 01483 683024. E-mail address: [email protected] (D.J. Ratliff).

http://dx.doi.org/10.1016/j.physd.2016.01.003 0167-2789/© 2016 Elsevier B.V. All rights reserved.

this case morph into the two-way Boussinesq equation. To show this, the strategy is to start with a Lagrangian formulation, average, and modulate. However when key derivatives with respect to wavenumber or frequency of the wave action and wave action flux are zero, higher derivatives are required and a slower time scale is needed. This combination changes the modulation equations from the dispersionless WMEs to the two-way Boussinesq equation which has dispersion. This result gives a new mechanism for the emergence of the two-way Boussinesq equation as a model PDE and, since the basic state can be a finite amplitude travelling wave or more generally a relative equilibrium, this observation increases the range of contexts in which it can appear. The 1 + 1 two-way Boussinesq equation in standard form is utt + u2 ± uxx



 xx

= 0,

(1.1)

for the scalar-valued function u(x, t ), with the plus (minus) sign corresponding to the good (bad) versions. This classification indicates whether the dispersion in the linear part, utt ± uxxxx , generates a well-posed (good) or ill-posed (bad) linear partial differential equation.

108

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

The first appearance of the Boussinesq equation was in equation (26) on page 75 of Boussinesq [5], where it was proposed as a model for water waves, and it has been extensively studied since: it has solitary wave solutions, blow-up for some initial data, and global existence for other initial data, and it is known to be completely integrable (e.g. [6–10]). The Whitham modulation theory [1,2] starts with a Lagrangian formulation of field equations for some vector of unknowns, say U (x, t ), L (U ) =



L(Ut , Ux , U ) dxdt ,

(1.2)

where for simplicity of discussion the field variables are restricted to (x, t ). Suppose there exists a periodic travelling wave solution U (x, t ) =  U (θ ),

 U (θ + 2π ) =  U (θ ), θ = kx + ωt + θ0 ,

(1.3)

of wavenumber k and frequency ω, where θ0 is an arbitrary constant phase shift. Average the Lagrangian over θ L (ω, k) =

1

2π



2π

L(ω Uθ , k Uθ ,  U ) dθ .

(1.4)

0

Now suppose ω and k are slowly varying in some sense and can be expressed in terms of the variance of the phase, k = θx

and ω = θt .

(1.5)

Combining these two equations gives the conservation of waves kt = ωx . Substituting (1.5) into (1.4) and taking variations with respect to θ of the averaged Lagrangian gives conservation of wave action At + Bx = 0 with A = Lω and B = Lk . Combining conservation of waves with conservation of wave action then gives the WMEs kt = ωx

and A (ω, k)t + B (ω, k)x = 0,

(1.6)

for the modulation variables (ω, k). The definition of wave action A and wave action flux B and the conservation of waves equation are all exact, but their validity depends on some approximation. Using multiple scales in the Whitham theory [11], slow time and space scales are introduced, T = ε t and X = ε x, and (ω, k) are taken in a neighborhood of some fixed travelling wave parameters,

ω → ω + ε Ω (X , T , ε) and k → k + ε q(X , T , ε). Then to leading order, q := q(X , T , 0) and Ω := Ω (X , T , 0) satisfy qT = ΩX

Aω ΩT + Ak qT + Bω ΩX + Bk qX = 0.

and

(1.7)

With (ω, k) fixed, this equation is linear and, with the assumption Aω ̸= 0, it can be written in the standard form

  q

Ω

+ A(ω, k)

  q

=

Ω

T

 

X

0 , 0

(1.8)

A(ω, k) =

1 Aω



−Aω . Ak + Bω

0 Bk

(1.9)

The WMEs are hyperbolic if the eigenvalues of A(ω, k) are real and elliptic if the eigenvalues of A(ω, k) are complex. Ellipticity is an indication that the basic state is unstable to long wave modulational instability [2]. The roots of det[λI − A(ω, k)] = 0 satisfy

λ=

Ak + Bω

2Aω



L = det ωω Lkω

±

1 √ Aω



−∆,

Lω k . Lkk

∆ = det

Aω ̸= 0

but Ak = Bk = 0,

(1.10)

Aω = Bω = 0.

(1.11)

or Bk ̸= 0

but

The main result of this paper is that in the neighborhood of the first singularity (1.10), the conservation of wave action morphs into the following form of the two-way Boussinesq equation qT = ΩX Aω ΩT + Bkk qqX + K qXXX = 0.

(1.12)

The first equation is just conservation of waves, and the second equation is the conservation of wave action. Combining the two equations in (1.12) gives the two-way Boussinesq equation in universal form

 Aω qTT +

1 2

2



= 0,

Bkk q + K qXX



Aω Bω

Ak Bk



(1.13)

XX

which we call the q-Boussinesq equation. Eq. (1.13) is universal in the sense that the coefficients are determined from abstract properties of a Lagrangian, and do not depend on particular equations. The second condition in (1.11) leads to

ΩX = qT Bk qX + Aωω ΩΩT + M ΩTTT = 0.

with



This form for the eigenvalues highlights the Lighthill condition [12]: when ∆ > 0 the WMEs are elliptic and when ∆ < 0 the WMEs are hyperbolic. The WMEs have been generalized in several directions: proper nonlinear WMEs, multiphase wavetrains, additional conservation laws, coupling with mean flow, extension to 2 + 1, and inclusion of dissipation, but attention here is restricted to the basic 1 + 1 singlephase conservative case to capture the nature of the singularityinduced appearance of the classical Boussinesq equation. The theory is then generalized to the 2 + 1 case which will lead to the WMEs morphing into the 2 + 1 Boussinesq equation. There has been a largely independent development of modulation theory for periodic solutions of non-conservative systems (Howard–Kopell, Burgers’ equation, Cross–Newell theory, Kuramoto–Sivashinsky, phase-diffusion equation, etc.). Conservation of waves is still operational in the non-conservative case, but conservation of wave action no longer exists in general, and new scaling is often needed, leading to a different class of modulation equations. For example, the generic phase equation for modulation of periodic wavetrains of reaction–diffusion equations is Burgers’ equation (e.g. Doelman et al. [13]) which requires the scaling with T = ε 2 t and X = ε x. Here we are interested in the case where the Whitham equations break down via a double zero eigenvalue of A(ω, k) in (1.8)–(1.9). We will also find that an additional natural requirement for the Boussinesq equation to emerge is that the double zero eigenvalue of A(ω, k) is non-semisimple. Because of the special form of A(ω, k) in (1.9) a double nonsemisimple zero eigenvalue can arise in one of two ways

(1.14)

Combining the two equations in (1.14) gives Bk ΩXX +



1 2

Aωω Ω 2 + M ΩTT



= 0,

(1.15)

TT

which we call the Ω -Boussinesq equation. The coefficients K and M are determined by a Jordan chain argument. Examples are given in Sections 7 and 8. The scaling in the first equation (1.13) is X = ε x and T = ε 2 t with scaling T = ε t and X = ε 2 x in the second equation (1.15). The theory extends in a natural way to the 2 + 1 case. Conservation of wave action has three components (A , B , C ), conservation of waves is vector valued, and θ in (1.3) is replaced by

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

θ = kx + my +ωt +θ0 . In the 2 + 1 case, the generic WMEs have four components and an additional slow variable Y . However, by adding a condition on Ck to (1.10), and modifying the transverse slow variable to Y = ε 2 y, the WMEs morph into a generalization of (1.13),  Aω qTT + 2Cω qYT + Cm qYY +

1 2

2



= 0,

Bkk q + K qXX

(1.16)

XX

with the condition Aω ̸= 0

but

Ak = Bk = Ck = 0,

(1.17)

with the generalization of (1.11) following in a similar way. Note the additional qYT term which arises naturally in the modulation theory and makes sense from a scaling perspective but does not appear in the usual 2 + 1 Boussinesq theory (e.g. [7]). On the other hand, in some examples Cω = 0 and so the cross derivative term vanishes (this happens in the example in Section 7). For simplicity much of the theory will be presented in the 1 + 1 case, with special details for the 2 + 1 case recorded as appropriate. The principal example is the complex Klein–Gordon (CKG) equation

  ψtt = ψxx + ψyy + a 1 − |ψ|2 ψ,

a = ±1,

(1.18)

for the complex-valued function ψ(x, y, t ). The CKG equation has a family of explicit periodic travelling wave solutions. Along particular branches of these solutions the singularities (1.10), (1.11) and (1.17) can occur. It is shown that near these singularities the modulation theory shows that the q-Boussinesq equation (1.13) emerges (when a = +1) and the Ω -Boussinesq equation emerges (when a = −1), as well as their generalizations to 2 + 1. In summary, it will be argued that when a branch of periodic travelling waves – of any amplitude – has one of the singularities (1.10)–(1.11), or the generalized 2 + 1 conditions, dispersion and nonlinearity are generated and the dynamics is governed by a twoway Boussinesq equation, on a periodic background, to leading order in ε . An outline of the paper is as follows. The relevant setup involving the Lagrangian to multisymplectic transformation, wave action, averaging, and linearization is presented in Sections 2 and 3. The modulation ansatz and the attendant calculations are introduced in Sections 4 and 5. The details of the CKG example with a = +1 are given in Section 7. In Section 8 the derivation of the Ω -Boussinesq equation is discussed. The principle example showing emergence of Ω -Boussinesq is modulation of the periodic travelling wave solutions of (1.18) with a = −1. This CKG equation arises as a weakly nonlinear model near threshold in the Kelvin–Helmholtz problem [14,15]. In Section 9 the question of the two-way Boussinesq equation for water waves is considered from a new perspective. A new derivation from the perspective of this paper suggests that the two-way Boussinesq model is not valid for water waves in shallow water unless the depth is of order ε 2 and then the equation breaks down. Concluding remarks are presented in Section 10. 2. From Lagrangian to multisymplectic Hamiltonian The starting point for the theory is the class of nonlinear wave equations that are generated by a Lagrangian. It would be of interest to develop the theory directly in terms of the Lagrangian, as in the Whitham theory. However, at present it is not clear how to do this. The theory as presented here relies strongly on the symplectic structures. These symplectic structures are generated, in principle, by a Legendre transform. Although in applications the Legendre transform for PDEs can be difficult in some cases, and so a direct construction of the symplectic structures may be more effective (e.g. [16]). However, an understanding of how a family

109

of symplectic structures is generated is easiest to see by using a sequence of Legendre transforms to transform to a multisymplectic Hamiltonian system [17,18]. In this formulation the conservation of wave action is given a geometric formulation [19] 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 L (U ) =



L(Ut , Ux , U ) dxdt ,

(2.1)

where U (x, t ) is in general vector valued. Undertake a Legendre transform V = δ L/δ Ut , giving a Hamiltonian formulation L H (W ) =

 

1 2

 ⟨MWt , W ⟩ − H (Wx , W ) dxdt ,

(2.2)

with new coordinates W = (U , V ), and ⟨·, ·⟩ an appropriate inner product. 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 (Wx , 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. In the 1 + 1 case, the Legendre transformation in the x-direction is equivalent to Legendre transforming the Hamiltonian function. Let P = δ L/δ Wx , and introduce a second Legendre transform, giving a multisymplectic Hamiltonian formulation L S (Z ) =

 

1 2

 1 ⟨MZt , Z ⟩ + ⟨JZx , Z ⟩ − S (Z ) dxdt , 2

(2.3)

with new coordinates Z = (U , V , P ), 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 three parts: a new Hamiltonian function S (Z ) which does not contain any derivatives with respect to t or x, and two symplectic structures represented by the skew-symmetric matrices M and J. 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 between the equations and the key conservation law. This connection appears in a central way in the modulation theory. In the 2 + 1 case an additional Legendre transform is introduced which generalizes (2.3) by the addition of a term 12 ⟨KZy , Z ⟩ with K a skew-symmetric matrix. 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 Section 7. Whitham also proposes the idea of Legendre transform of the time direction (see §14.5 of [2]) and gives alternative formulae for the coefficients in the Whitham modulation theory in terms of the averaged Hamiltonian. However, he does not Legendre transform in all directions and does not use the symplectic structure in the time direction or identify the symplectic structures in the spatial directions, all of which are needed to give an explicit relationship between symmetry and the conservation laws in the modulation theory. 2.1. The Euler–Lagrange equation, averaging and wave action It is assumed henceforth that the Lagrangian is in the canonical multisymplectic form (2.3) or its generalization to 2 + 1 and therefore the Euler–Lagrange equation is MZt + JZx + KZy = ∇ S (Z ),

Z ∈ R2n ,

(2.4)

with n ≥ 2 where S (Z ) is the Hamiltonian function, M, J, and K are skew-symmetric operators. The standard inner product on R2n , denoted by ⟨·, ·⟩, will be used.

110

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

Now assume that there exists a periodic travelling value solution of x-wavelength 2π /k, y-wavelength 2π /m, and period 2π /ω of the form

with other derivatives following a similar pattern. The first equation of (3.2) shows that  Zθ is in the kernel of L. It is natural to assume that the kernel is no larger:

Z ( x, y , t ) =  Z (θ ),

Kernel(L) = span  Zθ }.

 Z (θ + 2π ) =  Z (θ ), θ = kx + my + ωt + θ0 ,

(2.5)

with arbitrary phase shift θ0 . For the modulation theory the dependence on k, m, ω will be made explicit:  Z (θ , k, m, ω). There is the usual assumption on existence and smoothness of this solution so that the necessary differentiation in θ , k and ω is meaningful. The basic state satisfies

ωM Zθ + kJ Zθ + mK Zθ = ∇ S ( Z ).

(2.6)

To get the components of the conservation law for wave action, average the Lagrangian, evaluated on the family of travelling waves, over θ , L (ω, k, m)

=

1 2π

2π





ω 2

0

k

m

2

2

⟨M Zθ ,  Z ⟩ + ⟨J Zθ ,  Z⟩ +

 ⟨K Zθ ,  Z ⟩ − S ( Z ) dθ ,

and differentiate with respect to ω, k, and m, giving A (ω, k, m) = Lω = B (ω, k, m) = Lk =

1 2 1 2 1

C (ω, k, m) = Lm =

2

⟨⟨J Zθ ,  Z ⟩⟩

(2.7)

⟨⟨K Zθ ,  Z ⟩⟩,

1 2π

2π



L Zθ = 0 and L Zθ = 0 and

L Zk = J Zθ L Zω = M Zθ .

Double Jordan chains are nonstandard and it would be of great interest to develop a general theory. However, here we will treat the two Jordan chains independently. The second Jordan chain is taken to be fixed at length two, and we look at conditions for the first Jordan chain to extend to length four, and vice versa. For inhomogeneous equations that arise in the modulation theory and the Jordan chain theory, a solvability condition will be needed. With the assumption (3.3) and the symmetry of L, the solvability condition for the inhomogeneous equation LW = F is is solvable if and only if ⟨⟨ Zθ , F ⟩⟩ = 0.

(3.4)

4. Modulating the basic state The proposed modulation ansatz is Z (x, y, t ) =  Z (θ + εφ, k + ε 2 q, m + ε 3 r , ω + ε 3 Ω )

+ ε W (θ , X , Y , T ),

⟨U , V ⟩ dθ .

(2.8)

0

Note that the structure matrices M, J, and K appear in the Euler–Lagrange equations, (2.4) and (2.6), and in the components of the conservation law (2.7). Key derivatives that appear as coefficients in the WMT are Aω = ⟨⟨M Zθ ,  Zω ⟩⟩,

(3.3)

The second and third equations of (3.2) shows that there are nontrivial Jordan chains associated with the zero eigenvalue of L with geometric eigenvector  Zθ . There are in fact two Jordan chains of length two in (3.2)

LW = F

⟨⟨M Zθ ,  Z ⟩⟩

where ⟨⟨·, ·⟩⟩ is the inner product averaged over θ ,

⟨⟨U , V ⟩⟩ :=



Ak = ⟨⟨M Zθ ,  Zk ⟩⟩,

Am = ⟨⟨M Zθ ,  Zm ⟩⟩,

Bω = ⟨⟨J Zθ ,  Zω ⟩⟩,

Bk = ⟨⟨J Zθ ,  Zk ⟩⟩,

Bm = ⟨⟨J Zθ ,  Zm ⟩⟩, (2.9)

Cω = ⟨⟨K Zθ ,  Zω ⟩⟩,

Ck = ⟨⟨K Zθ ,  Zk ⟩⟩,

Cm = ⟨⟨K Zθ ,  Zm ⟩⟩.

(4.1)

where φ, q, r and Ω are all functions of the slow variables X , Y and T, X = ε x,

Y = ε 2 y,

T = ε2 t ,

and

(4.2)

and  Z (θ , k, m, ω) is the basic state. W represents a correction term to account for the fact that modulation of  Z alone may not satisfy the governing equations exactly. Expand W in a Taylor series

ε W = ε W1 + ε 2 W2 + ε 3 W3 + ε 4 W4 + ε 5 W5 + O (ε6 ).

(4.3)

Bkk = ⟨⟨J Zθ ,  Zkk ⟩⟩ + ⟨⟨J Zθ k ,  Zk ⟩⟩.

It becomes apparent a posteriori that W1 and W2 can be taken to be zero, so the first interesting term in this expansion is W3 . The appropriate scaling for q, r and Ω is dictated by their definition in terms of the phase function,

The definitions (2.7) give the cross-derivatives

q = φX ,

There is also the appearance of the second k derivative in B appearing at the end of the analysis, taking the form

Bω = Lkω = Lωk = Ak ,

Ck = Lmk = Lkm = Bm ,

Cω = Lmω = Lωm = Am .

and (2.10)

3. Linearization about the periodic basic state Define the linear operator



Lf = D2 S ( Z ) − kJ

d dθ

− mK

d dθ

− ωM

d dθ



f,

(3.1)

obtained by linearizing (2.6). Then differentiating (2.6), D2 S ( Z ) Zθ = kJ Zθθ + ωM Zθθ , D2 S ( Z ) Zk = kJ Zθ k + ωM Zθ k + J Zθ , or L Zθ = 0,

L Zk = J Zθ ,

and L Zω = M Zθ ,

(3.2)

r = φY ,

and

Ω = φT ,

(4.4)

and they give the conservation of waves by cross differentiation qT = ΩX ,

qY = rX ,

and

rT = ΩY .

(4.5)

The strategy is to expand the modulation ansatz (4.1) 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 measures the distance from criticality in k space. Therefore if k0 is a value of k satisfying Bk = 0, then k − k0 = ε 2 q with q of order one. The 2 + 1 case does not involve great additional detail, and so both the 1 + 1 and 2 + 1 cases are considered together here. Also the details will be given for the first singularity (1.10) and then the details for the dual case (1.11) are summarized in Section 8. Taylor expanding the modulation of the basic state, we can write

 Z (θ + εφ, k + ε 2 q, m + ε 3 r , ω + ε 3 Ω ) =

5  n=0

ε n Zn + O (ε6 )

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

When J is invertible then this chain is a classical Jordan chain. However, in the present case, J may not be invertible. Hence we include the assumption

with Z0 =  Z (θ , k, m, ω), Z3 = Z4 = Z5 =

1

Z1 = φ Zθ ,

Z2 =

1 2

φ 2 Zθθ + q Zk

ξi ̸∈ Ker(J),

φ 3 Zθθθ + qφ Zθ k + r Zm + Ω Zω ,

6 1

120 1

2

2

1

1 1 φ 5 Zθθθθθ + qφ 3 Zθθθ k + q2 φ Zθ kk + φ 2 r Zθ θ m 6

2

i = 1, . . . , J ,

(4.12)

which appears to be satisfied in examples. Here we are interested in the Jordan chain associated with the geometric eigenvector ξ1 =  Zθ . As shown in (3.2) the Jordan chain associated with  Zθ has length at least two since ξ2 =  Zk . In fact it has length at least three due to (4.10). It has length four if Lξ4 = Jξ3 and this equation is solvable if and only if

1 1 Zθθθθ + qφ 2 Zθθ k + q2 Zkk + φ r Zθ m + φ Ω Zθ ω φ 4

24 1

111

2

+ φ Ω Zθθω + qr Zkm + qΩ Zkω , 2

2

with each term in the expansion evaluated at θ , k, m, ω. Full expansions of all the terms involved in the modulation form rather lengthy expressions, therefore just a summary of the key steps is given here. The zeroth order equation recovers the governing equation for  Z . The first order equation gives φ L Zθ = 0 which is satisfied exactly. The second order equation is satisfied exactly due to (4.4).

⟨⟨ Zθ , Jξ3 ⟩⟩ = 0.

(4.13)

However,

⟨⟨ Zθ , Jξ3 ⟩⟩ = −⟨⟨J Zθ , ξ3 ⟩⟩ = −⟨⟨L Zk , ξ3 ⟩⟩   = −⟨⟨Zk , Lξ3 ⟩⟩ = −⟨⟨Zk , J Zk ⟩⟩ = 0,

with the last equality due to skew symmetry of J. Hence the Jordan chain has length at least four. There is no fifth element if

⟨⟨ξ1 , Jξ4 ⟩⟩ ̸= 0.

4.1. Third order terms

(4.14)

(4.15)

Assume this condition is satisfied and define K = ⟨⟨Jξ1 , ξ4 ⟩⟩ ̸= 0.

The problem at third order reads

 1 3 φ L Zθθθ + 3D3 S ( Z )( Zθθ ,  Zθ ) + D4 S ( Z )( Zθ ,  Zθ ,  Zθ ) 6   + qφ L Zθ k + D3 S ( Z )( Zk ,  Zθ ) − J Zθθ + (r − φY )L Zm + (Ω − φT )L Zω + LW3 − qX J Zk = 0 .

It is precisely this coefficient that arises in the modulation theory to give the coefficient of dispersion in the x-direction. 4.3. Fourth order terms (4.6)

Terms proportional to φ 3 and qφ can be shown to vanish identically, since one can verify by differentiating (2.6) that

− qY (J Zm + K Zk ) − qXX Jξ3 = 0.

and L Zθ k = J Zθθ − DS ( Z )( Zk ,  Zθ ). Many terms in the analysis at each order cancel in this way, so in light of this we will instead omit such terms from expressions at each order. This leaves (4.7)

Applying (4.4), the first two terms vanish. The remaining system, LW3 = qX J Zk , is solvable if

⟨⟨ Zθ , J Zk ⟩⟩ = −⟨⟨J Zθ , Zˆk ⟩⟩ = −Bk = 0,

(4.8)

and so we require that B is k-critical in order to continue with the analysis. This solvability condition confirms the latter condition in (1.10). The solution for W3 is then W3 = α(X , Y , T ) Zθ + qX ξ3 ,

(4.9)

for some arbitrary function α , and ξ3 is defined through the relation Lξ3 = J Zk .

After various simplifications using relations already determined, we are left at fourth order with the equation L(W4 − αX Zk − αφ Zθ θ − qX φ(ξ3 )θ ) − qT (J Zω + M Zk )

L Zθ θ θ = −3D3 S ( Z )( Zθθ ,  Zθ ) − D4 S ( Z )( Zθ ,  Zθ ,  Zθ ),

(r − φY )L Zm + (Ω − φT )L Zω + LW3 − qX J Zk = 0 .

(4.16)

(4.10)

This equation is solvable, and so the solution ξ3 exists, under the condition Bk = 0.

(4.17)

In the above, the term prefactored by qX φ being absorbed into the linear operator is a consequence of differentiating (4.10) with respect to θ : D2 S ( Z )(ξ3 )θ + D3 S ( Z )( Zθ , ξ3 ) − (ωM + kJ + mK)(ξ3 )θ = J Zθ k 3    ⇒ L(ξ3 )θ = JZθ k − D S (Z )(Zθ , ξ3 ). Checking the solvability of the qY term in (4.17),

⟨⟨ Zθ , J Zm + K Zk ⟩⟩ = −⟨⟨J Zθ ,  Zm ⟩⟩ + ⟨⟨ Zθ , K Zk ⟩⟩ = −⟨⟨L Zk ,  Zm ⟩⟩ + ⟨⟨ Zθ , K Zk ⟩⟩ = −⟨⟨ Zk , L Zm ⟩⟩ + ⟨⟨ Zθ , K Zk ⟩⟩       = −⟨⟨Zk , KZθ ⟩⟩ + ⟨⟨Zθ , KZk ⟩⟩ = 2⟨⟨Zθ , KZk ⟩⟩ = −2Ck . Thus, it is solvable only if Ck = 0 .

(4.18)

The second component of the wave action flux C needs to be extremal with respect to k to continue. A similar calculation of solvability for the qT term in (4.17) imposes that 0 = ⟨⟨ Zθ , J Zω + M Zk ⟩⟩ = −2Ak . In summary the equations are solvable up to fourth order only if Ak = Bk = Ck = 0.

This leads to a solution for W4 at this order as W4 = αX Zk + αφ Zθ θ + β(X , Y , T ) Zθ + qX φ(ξ3 )θ

4.2. Interlude—Jordan chains

+ qY η + qT ζ + qXX ξ4 ,

A Jordan chain in J of length J, {ξ1 , . . . , ξJ }, for a zero eigenvalue in the J-symplectic setting is defined by

with η and ζ defined by

Lξ1 = 0,

Lη = J Zm + K Zk ,

Lξi = Jξi−1 ,

i = 2, . . . , J .

(4.11)

Lζ = J Zω + M Zk .

(4.19)

(4.20)

112

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

5. Fifth order and the Boussinesq equation At fifth order the solvability condition generates the conservation of wave action relative to the new scaling. Combining this form with conservation of waves will lead to the two way Boussinesq equation. The fifth order terms obtained from the substitution of (4.1) into the governing equations is lengthy. However, after simplification it can be reduced to the manageable equation,

Now evaluating the coefficients in turn. The coefficient of the hydrodynamic term, qqX , can be seen to be prefactored by the negative second k derivative of B since Bkk = ⟨⟨J Zθ ,  Zkk ⟩⟩ + ⟨⟨J Zθ k ,  Zk ⟩⟩ = ⟨⟨J Zθ ,  Zkk ⟩⟩ − ⟨⟨ Zθ k , J Zk ⟩⟩

= ⟨⟨J Zθ ,  Zkk ⟩⟩ − ⟨⟨ Zθ k , Lξ3 ⟩⟩ = ⟨⟨J Zθ ,  Zkk ⟩⟩ − ⟨⟨L Zθ k , ξ3 ⟩⟩ 3       = ⟨⟨JZθ , Zkk ⟩⟩ − ⟨⟨JZθ θ − D S (Z )(Zk , Zθ ), ξ3 ⟩⟩ = −⟨⟨ Zθ , J Zkk ⟩⟩ − ⟨⟨J Zθ θ , ξ3 ⟩⟩ + ⟨⟨D3 S ( Z )( Zk ,  Zθ ), ξ3 ⟩⟩ 3 = −⟨⟨ Zθ , J Zkk ⟩⟩ − ⟨⟨ Zθ , J(ξ3 )θ ⟩⟩ + ⟨⟨ Zθ , D S ( Z )( Zk , ξ3 )⟩⟩. (5.8)

L W5 − φ 2 α Zθθθ − qα Zθ k − βφ Zθθ − αXX ξ3



 − βX Zk − φ qT (η)θ − φ qXX (ξ4 )θ − qXT (Jζ + Mξ3 ) − qXY (Jη + Kξ3 ) − qqX (J Zkk + J(ξ3 )θ − D3 S ( Z )( Zk , ξ3 )) − rY KZˆm − rT MZˆm − qXXX Jξ4 − ΩT M Zω − ΩY K Zω = 0.

The coefficient a3 can be calculated as

−⟨⟨Zˆθ , MZˆm + K Zω ⟩⟩ = Am + Cω ≡ 2Cω , (5.1)

Define W5 − φ 2 α Zθθθ − qα Zθ k − βφ Zθθ − αXX ξ3 − βX Zk Then (5.1) simplifies to

5 − qXT (Jζ + Mξ3 ) − qXY (Jη + Kξ3 ) − qqX (J LW Zkk + J(ξ3 )θ − D3 S ( Z )( Zk , ξ3 )) − rY KZˆm − rT MZˆm − qXXX Jξ4 − ΩT M Zω − ΩY K Zω = 0.

(5.2)

This equation can be simplified further by noting that Jζ + Mξ3 is in the range of L. This follows since

+ Cm rY + 2Cω rT = 0.

(5.10)

X

This remarkable equation is conservation of wave action with respect to the new scaling. Differentiation of this equation with respect to X leads to (1.16). In general the modulation functions, q, r , φ, Ω , depend on ε , and the above result gives the leading order q ≡ q(X , Y , T , 0), etc. In general, φ, q, r and Ω can be expanded in a Taylor series in ε and the analysis can, in principle, be continued to arbitrary order.

E= (5.3)

A similar calculation showing that the Jη + Kξ3 is also in the range of L. Therefore there exists a function η2 satisfying (5.4)

Hence (5.2) simplifies further to

5 − qXT ζ2 − qXY η2 = qqX (J L W Zkk + J(ξ3 )θ − D3 S ( Z )( Zk , ξ3 )) 

(5.5)

Imposing solvability then leads to an equation of the form a0 ΩT + a1 qqX + a2 rY + a3 rT + a4 qXXX = 0,



(6.1)

with

Therefore exists a function ζ2 satisfying

+ rY KZˆm + rT MZˆm + qXXX Jξ4 + ΩT M Zω + ΩY K Zω .

2

Bkk q2 + K qXX

ET + F X = 0 ,

= ⟨⟨ξ3 , MZˆθ ⟩⟩ + ⟨⟨Zˆθ , Mξ3 ⟩⟩ = 0.



1

The Boussinesq equation (1.12) has an energy conservation law. In the 1 + 1 case it is

= ⟨⟨JZˆk , Zˆω ⟩⟩ + ⟨⟨Zˆθ , Mξ3 ⟩⟩ = ⟨⟨Lξ3 , Zˆω ⟩⟩ + ⟨⟨Zˆθ , Mξ3 ⟩⟩ = ⟨⟨ξ3 , Zˆω ⟩⟩ + ⟨⟨Zˆθ , Mξ3 ⟩⟩

Lη2 = Jη + Kξ3 .



6. An energy equation

⟨⟨Zˆθ , Jζ + Mξ3 ⟩⟩ = −⟨⟨LZˆk , ζ ⟩⟩ + ⟨⟨Zˆθ , Mξ3 ⟩⟩ = −⟨⟨Zˆk , JZˆω ⟩⟩ − ⟨⟨Zˆk , MZˆk ⟩⟩ + ⟨⟨Zˆθ , Mξ3 ⟩⟩

Lζ2 = Jζ + Mξ3 .

using the third set of cross derivatives in (2.10). Collecting these results, for the ansatz (4.1) to satisfy the governing equation up to fifth order we must have that Aω ΩT +

5 . − φ qT (η)θ − φ qXX (ξ4 )θ = W

(5.9)

(5.6)

using rT = ΩY from the conservation of waves. It remains to show that these coefficients can be expressed in terms of derivatives of the components of the conservation of wave action. Explicitly, these coefficients are a0 = −⟨⟨ Zθ , M Zω ⟩⟩ = ⟨⟨M Zθ ,  Zω ⟩⟩ = Aω

1 2

Aω Ω 2 +

1 2

K q2X −

1 6

Bkk q3 ,

and

 F = K det

Ω

ΩX

qX

qXX



1

+ Bkk Ω q2 . 2

(6.3)

This conservation law can be verified by direct calculation. One of the key features that this conservation law highlights is that the energy E is not sign definite in general, because of the cubic term, even when the Boussinesq equation (1.13) is dispersively wellposed (Aω K > 0). Hence the nonlinearity can be a further obstacle to global existence. See Bona & Sachs [9] for further discussion of the implications of energy estimates in the two-way Boussinesq equation. A similar construction gives an energy equation for the Ω -Boussinesq equation (1.14). 7. Example: complex Klein–Gordon equation Consider the complex Klein–Gordon (CKG) equation with cubic nonlinearity,

ψtt = ψxx + ψyy + ψ − |ψ|2 ψ,

   a1 = −  Zθ , J Zkk + J(ξ3 )θ − D3 S ( Z )( Zk , ξ3 )

(6.2)

(7.1)

a4 = −⟨⟨ Zθ , Jξ4 ⟩⟩ = ⟨⟨Jξ1 , ξ4 ⟩⟩ = K ,

where ψ(x, y, t ) is a complex valued function. The CKG equation has an SO(2) symmetry in that eiθ ψ(x, y, t ) is a solution for any θ when ψ(x, y, t ) is a solution. Associated with this symmetry, the CKG equation has a class of exact periodic nonlinear travelling wave solutions,

where (4.16) has been used in the last equation.

ψ(x, y, t ) = Ψ0 eiθ ,

a2 = −⟨⟨ Zθ , K Zm ⟩⟩ = ⟨⟨K Zθ , Zˆm ⟩⟩ = Cm a3 = −⟨⟨ Zθ , MZˆm + K Zω ⟩⟩

(5.7)

θ = kx + my + ωt + θ0 .

(7.2)

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

113

It will be shown below that the coefficient of dispersion is K = − 21 . Hence, according to the theory of this paper the emergent Boussinesq equation for ε sufficiently small is 2 3

 qTT +

√ 1 ± 3q2 + qXX



2

− qYY = 0

2

3

XX

or

√

 qTT +

Fig. 1. Dispersion relation and branch of periodic travelling waves showing the codimension two point where dispersive Boussinesq dynamics emerges. The horizontal axis is k, the vertical is |A| and the third axis is ω.

±

3 3 2

3

2

q +

4

 − qYY = 0,

qXX

(7.8)

XX

which is the 2 + 1 good Boussinesq equation (in the sense that the linear part qTT + 12 qXXXX − qYY is well posed). As far as we are aware, this is the first time that it has been shown that the CKG can be reduced to the two-way Boussinesq equation along a branch of finite-amplitude travelling waves. When Bk = Ck = 0 but Ak ̸= 0 then KP emerges [20] on the slower KdV time scale, but when all the first derivatives of (A , B , C ) are nonzero, then the standard WMEs emerge on the faster time scale ε t. To show that the above reduction of CKG to 2 + 1 Boussinesq fits into the theory in this paper we need to show that the equation has a Lagrangian and multisymplectic formulation, and verify the value of the coefficient of dispersion K .

Substitution of this form into (7.1) shows that Ψ0 satisfies

|Ψ0 |2 = ω2 − k2 − m2 + 1.

(7.3)

The components of the conservation law associated with the SO(2) symmetry are A = −ℑ(ψ ∗ ψt ),

B = ℑ(ψ ∗ ψx ),

and

7.1. Basic state in multisymplectic coordinates The CKG equation (7.1) is the Euler–Lagrange equation associated with the Lagrangian density

C = ℑ(ψ ∗ ψy ). (7.4)

L=

It can be verified by direct calculation that At + Bx + Cy = 0, when ψ is a solution of (7.1). Evaluate the components of the conservation law along the branch of solutions (7.2)–(7.3) A (k, m, ω) = −ω|Ψ0 |2 = ω(k2 + m2 − ω2 − 1) B (k, m, ω) = k|Ψ0 |2 = k(1 + ω2 − k2 − m2 ) C (k, m, ω) = m|Ψ0 |2 = m(1 + ω2 − k2 − m2 ).

1



2

 1 |ψx |2 + |ψy |2 − |ψ|2 − |ψt |2 + |ψ|4 . 2

Introduce real coordinates ψ = a1 + ia2 ,

L=

(7.5)

1



2

−(

∂ a1 ∂x

a21

+

2

 +

a22

1

∂ a2 ∂x

)+ ( 2

a21

2

+

 + )

a22 2



∂ a1 ∂y

2

 +

∂ a2 ∂y

2

 −

∂ a1 ∂t

2

.

Compute the derivatives that are needed for the necessary condition (1.17)

To transform to canonical multisymplectic form, introduce a sequence of Legendre transformations, namely

Ak = 2kω,

∂L ∂ ai = , ∂(∂x ai ) ∂x ∂L ∂ ai di = =− . ∂(∂t ai ) ∂t

Bk = 1 − 3k2 − m2 + ω2 ,

Ck = −2km.

bi =

The only nontrivial solution of these three equations is

ω = m = 0 and 3k2 = 1 ⇒ |Ψ0 |2 = 1 − k2 =

2

.

(7.6)

3 The curve of periodic solutions and their singularities is shown in Fig. 1. The linear dispersion relation is ω2 = k2 + m2 − 1 and when m = 0, which will be required for the emergence of Boussinesq, it is ω2 = k2 − 1, and the two branches of that curve are shown as the green curves in Fig. 1. The linear normal modes are unstable in the region −1 < k < +1. The branch of nonlinear periodic travelling waves, noted in red in the figure, connects the two branches of the dispersion relation and exists precisely for the wavenumbers that are linearly unstable. Along that branch there are exactly two codimension two points (m = 0 and k2 = 13 ). It is these two points that are organizing centers for the emergence of dispersive Boussinesq dynamics. Now compute the other coefficients that appear in (1.16), 2

√

Aω = k2 − 3ω2 − 1 = k2 − 1 = − , Bkk = −6k = ∓2 3 3 (7.7) 2 Cm = 1 + ω2 − 3m2 − k2 = , Cω = 2ωm = 0. 3

ci =

∂L ∂ ai = , ∂(∂y ai ) ∂y

(7.9)

Hence, with new coordinates Z = (a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 ), the CKG-equation can be written in canonical multi-symplectic form (2.4) with S (Z ) =

1 2

 a·a+b·b+c·c−d·d−

1 2

 (a · a)2 ,

and



0 0 M= 0 I



0 0 K= I 0

0 0 0 0

0 0 0 0

0 0 0 0

−I 0 0 0

−I



0 , 0 0



0 0 . 0 0



0 I J= 0 0

−I 0 0 0

0 0 0 0



0 0 , 0 0

114

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

In multisymplectic coordinates, the basic state is Rθ  u  kσ Rθ  u  ,  Z (θ , k, m, ω) =   mσ Rθ  u  −ωσ Rθ  u 1 σ = R− θ

d dθ

q-Boussinesq can be redone by switching x and t and switching the role of the components of wave action, leading to conditions for the emergence of Ω -Boussinesq. Take the conditions (1.11) as a starting point





 Rθ =

cos θ sin θ

 − sin θ , cos θ

(7.10)

X = ε2 x

7.2. K -dispersion calculation

u (1 − a · a)u − 2(u · a)a + ωzθ + kvθ + mwθ v − kuθ  v  L  =  . w − muθ w −z − ωuθ z





The Jordan chain satisfies Lξj = Jξj−1 for j = 1, 2, 3, 4 with ξ0 = 0 and ξ1 =  Zθ with  Z given in (7.10). Since all terms will have a factor Rθ , express the Jordan chain elements in the form Rθ αj Rθ βj  ξj =  , Rθ γ j  Rθ δj



= = = =

−βj−1 − kσαj−1 , kσαj + αj−1 , mσαj

(7.11)

−ωσαj ,

for j = 1, 2, 3, 4, with initial vectors α0 = β0 = γ 0 = δ0 = 0. Solving gives the Jordan chain

 σ Rθ  u u  −kRθ  , ξ1 =  −mRθ  u ω Rθ  u   





Rθ  u 1 −kσ Rθ  u ξ2 = −  , mσ Rθ  u 2k −ωRθ  u



0 1 Rθ  u ξ3 = −   , 0 2k 0

3

1

(k∥ u∥2 + k∥σ u∥2 ) = − . 2

= 0,

(8.2)

where M is determined by a Jordan chain argument associated with the degeneracy Aω = 0. This version of the Boussinesq equation is less interesting as the initial-value problem is unstable. However, the singularity Aω = Bω = 0 does appear in applications. 8.1. Kelvin–Helmholtz instability and Ω -Boussinesq Near the Kelvin–Helmholtz (KH) instability, a complex Klein– Gordon equation can be derived in the form (8.3)

(e.g. [14] and [15]), with obvious generalization to 2 + 1. This equation is similar to the 1 + 1 version of (7.1) with a sign change in the last two terms. In this case the dispersion relation of the linear problem is stable: ω2 = k2 + 1. Eq. (8.3) applies to the region just before the KH instability, with −ψ replaced by +ψ in the KH unstable region. There still exists a branch of nonlinear solutions of the form (7.2) with

|Ψ0 |2 = 1 + k2 − ω2 ,

(8.4)

with the condition ω < 1 + k . The conservation law is the same as (7.4) reduced to 1 + 1, Evaluate the components of the conservation law along the branch of solutions (8.4) 2

2

(8.5)

Bω = −2ωk,

Bk = 1 + 3k2 − ω2 .

The only non-trivial solution of the first two conditions is k = 0 and 3ω2 = 1 and at these values Bk ̸= 0. Since Aωω ̸= 0, the form of the Ω -Boussinesq equation is (7.12)

8. Dual Boussinesq equation and the KH instability There is a dual version of the above theory when space and time are reversed. In this section the 1 + 1 reduction to Ω Boussinesq equation will be discussed, with the generalization to any space dimension following similar lines. In the 1 + 1 case the Euler–Lagrange equation (2.4) simplifies to Z ∈ R2n .

 TT

Aω = −1 − k2 + 3ω2 ,

This completes the verification of (7.8).

MZt + JZx = ∇ S (Z ),

2

Aωω Ω 2 + M ΩTT

and compute the derivatives needed for the necessary condition (1.17)



Rθ  u 3  kσ Rθ  u  ξ4 = −  . mσ Rθ  u  8k  −ωσ Rθ u

8k

1

A (k, ω) = −ω|Ψ0 |2 = −ω(1 + k2 − ω2 ) B (k, ω) = k|Ψ0 |2 = k(1 + k2 − ω2 ),

Hence K = ⟨⟨Jξ1 , ξ4 ⟩⟩ = −



ψtt = ψxx − ψ + |ψ|2 ψ,

j = 1, . . . , 4.

Substitution into the Jordan chain equation gives the following sequence for the components of each element in the Jordan chain

−2(αj ·  u) u βj γj δj

and T = ε t ,

and switch the scalings on Ω and q in order to balance conservation of waves. Then re-doing the above argument in Sections 4–5 shows that the WMEs morph into the Ω -Boussinesq equation Bk ΩXX +

To compute the K -dispersion for this system, we first need to set up the linearization L, defined in (3.1) and identify the Jordan chain. The linearization is



Bk ̸= 0.

but

Switch the scalings to

Rθ ,

with  u · u = 1 − k2 − m2 + ω2 . With the conditions (7.6) this simplifies to ∥ u∥2 = 23 .

 

A ω = Bω = 0

(8.1)

One advantage of the multisymplectic formulation is that space and time are on an equal footing, and so the argument leading to

 ΩXX + ±

√ 3 3 2

Ω + 2

3 2

 = 0.

M ΩTT TT

A calculation (see below in Section 8.2) shows that M = − 21 . Thus the overall Boussinesq equation stemming from the KH instability problem reads

 ΩXX + ±

√ 3 3 2

Ω − 2

3 4

 = 0.

ΩTT

(8.6)

TT

Although the resulting Ω -Boussinesq equation is ill-posed (as an evolution equation in time), it has many bounded solutions. For example, take

 (ξ ), Ω (X , T ) = a + Ω

ξ = T + bX ,

(8.7)

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

 (ξ ) a function satisfying Ω  (ξ ) → 0 as ξ → with a constant, and Ω ±∞. Substitution of the form (8.7) into (8.6) gives the following , equation for Ω √ 3 4

 − µΩ  = 0, Ω − (b + 2aµ)Ω  ′′

2

2

µ=±

3 3 2

fails. The analogue for uniform flows of action conservation is mass conservation with densities A ( u0 , r ) =

.

2

8.2. Calculation of M This is similar to the calculation of K in Section 7.2, and almost by inspection it can be seen that M = − 12 but we sketch the calculation strategy for completeness. For (8.3) take

 a=

Re(ψ) , Im(ψ)



b = ax ,

c = −at .

Then a 1 + 1 multisymplectic formulation of (8.3) is MZt + JZx = ∇ S (Z ), with a b , c

  Z =

0 0 −I

 M=

0 0 0

I 0 , 0



0 −I 0

 J=

I 0 0

0 0 , 0



1 M = ⟨⟨Mξ1 , ξ4 ⟩⟩ = − . 2

ηtt − gh0 ηxx = gh0

3 η2 2 h0

+

h20 3

ηxx



,

1 2

2

u20



and B (u0 , r ) =

u0 g

 r−

1 2

u20



,

u20 ,

0=

and

u0 ∂A =− . ∂ u0 g

The first condition gives Froude number criticality (u20 = gh0 ) but the second condition requires u0 = 0 and so h0 = 0. Hence the conclusion of the modulation theory is that the two-way Boussinesq equation is not a valid model for shallow water hydrodynamics. This conclusion is not surprising since the two-way Boussinesq equation can only arise if there is no preferred direction in the basic state. The derivation of KdV, KP, and two-way Boussinesq for water waves starts with either a quiescent fluid of uniform depth h0 and a moving frame, or a stationary frame with a background uniform velocity and depth (u0 , h0 ). With either a moving frame, or a nonzero background velocity u0 , a preferred direction is specified, and so it is not expected that a two-way Boussinesq equation will emerge. Going back to (9.1) it is clear there is going to be a problem with asymptotic validity. For example, introduce the Boussinesq scaling X = ε x,

η = ε 2 q,

into (9.1) qTT −

gh0

ε2

 qXX = gh0

3 q2 2 h0

+

h20 3



.

qXX

(9.2)

XX

This equation does not appear to be asymptotically valid unless h0 ∼ ε 2 which is consistent with what was found with the modulation argument. However, if h0 ∼ ε 2 then the dispersion term is of higher order, nullifying the two-way Boussinesq equation. Alternatively one can use a linear scaling X = ε x,

η = ε q,

which when substituted into (9.1) gives

The most well known context for the appearance of the twoway Boussinesq equation is in shallow water hydrodynamics. In the 1 + 1 case, it is found to be



g

1

r−

  ∂B 1 3 2 = r − u0 0= ∂ u0 g 2

T = εt ,

9. The two-way Boussinesq equation for water waves



where (u0 , r ) are the analogue of (ω, k) and represent the uniform velocity and Bernoulli constant respectively (cf. [3]). The analogue of the conditions (1.10) are

T = ε2 t ,

2

and S (Z ) = 12 a · a − 21 b · b + 12 c · c − 4 a · a . The strategy for computing M then follows the same strategy as in Section 7.2. The linearization L is set up, and then a sequence for the Jordan chain is constructed. We omit the details as they are similar to Section 7.2. The only difference is that now M is the key matrix in the definition of M ,

 1

1

r = gh0 +

With a chosen so that b + 2aµ > 0 and taking µ < 0 this equation has a sech2 solution. The solution (8.7) is just one example of many bounded solutions of (8.6). There are also periodic, quasi-periodic, and multi-pulse solutions which can be obtained by adapting the solutions in [21] (e.g. switching x and t). These solutions may be of physical interest near the Kelvin–Helmholtz threshold.

115

(9.1)

xx

where η(x, t ) is the perturbation free surface, h0 is the mean depth, and g is the gravitational constant. A derivation of this equation in the Eulerian setting is given in Keulegan & Patterson [22] (see §3 starting on page 72 of [22]). Ursell [23] derives the same equation in the Lagrangian particle path setting. It is extended to the 2 + 1 case, with the addition of a ηyy term, in Johnson [7]. In principle this two-way Boussinesq equation should follow from the theory of this paper by modulating the basic state in shallow water hydrodynamics—the uniform flow. In [3] it is shown that modulation of the uniform flow, at criticality, leads to emergence of the Korteweg–de Vries (KdV) equation, and in [24] modulation and criticality lead to the Kadomtsev–Petviashvili (KP) equation. Hence, with one additional elementary condition, the twoway Boussinesq equation should emerge. However, this argument

 qTT − gh0 qXX = gh0

3 q2

ε

2 h0

+ε

2 2 h0

3



.

qXX

(9.3)

XX

Hence to leading order it degenerates to a linear wave equation. In summary, there does not appear to be any asymptotic scaling with (9.1) as the leading order term. The inadequacies of the two-way Boussinesq equation have been noted in the literature before from other perspectives. For example, Keulegan & Patterson [22] argue that the derivation requires the imposition of initial conditions so that the Boussinesq equation is restricted to be a one-way equation (effectively reducing it to KdV). Specifically they state between equations (102) and (103) that ‘‘. . . restricting ourselves to waves propagated in the positive x-direction, we obtain . . . ’’ Thereby nullifying the two-way property. The rigorous validity theory of Schneider & Wayne [25,26] argue that the KdV equation is the only consistent 1 + 1 model approximating the full water-wave problem in shallow water. Specifically, in §5.2 of Schneider & Wayne [26] it is argued that the Boussinesq equation will not sustain a validity argument because of the presence of ε terms in the equation (as in (9.3)). More

116

D.J. Ratliff, T.J. Bridges / Physica D 333 (2016) 107–116

precisely, they argue in Schneider & Wayne [25] that the only longwave model which is asymptotically valid is the decoupled left and right running KdV equations. On the other hand, the two-way Boussinesq equation may be a valid model in other water-wave contexts, for example along a branch of periodic waves, without a preferred direction, with a singularity of the form (1.10).

validity of the two-way Boussinesq equation as a model for shallow water waves.

10. Concluding remarks

[1] G.B. Whitham, A general approach to linear and nonlinear dispersive waves using a Lagrangian, J. Fluid Mech. 22 (1965) 273–283. [2] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [3] T.J. Bridges, A universal form for the emergence of the Korteweg–de Vries equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013) 20120707. [4] T.J. Bridges, Breakdown of the Whitham modulation theory and the emergence of dispersion, Stud. Appl. Math. (2015) http://dx.doi.org/10.1111/sapm.12086. [5] J.V. Boussinesq, Théorie des ondes et des remous qui se propagent le long dun canal rectangulaire horizontal, en communiquant au liquide contene dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. (9) 17 (1872) 55–108. [6] L.V. Bogdanov, V.E. Zakharov, The Boussinesq equation revisited, Physica D 165 (2002) 137–162. [7] R.S. Johnson, A two-dimensional Boussinesq equation for water waves and some of its solutions, J. Fluid Mech. 323 (1996) 65–78. [8] C. Liu, Z. Dai, Exact perioidic solitary wave solutions for the (2 + 1)-dimensional Boussinesq equation, J. Math. Anal. Appl. 367 (2010) 444–450. [9] J.L. Bona, R.L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys. 118 (1988) 15–29. [10] S.K. Turitsyn, Blow-up in the Boussinesq equation, Phys. Rev. E 47 (1993) R796–R799. [11] G.B. Whitham, Two-timing, variational principles and waves, J. Fluid Mech. 44 (1970) 373–395. [12] M.J. Lighthill, Contribution to the theory of waves in non-linear dispersive systems, J. Inst. Math. Appl. 1 (1965) 269–306. [13] A. Doelman, B. Sandstede, A. Scheel, G. Schneider, The dynamics of modulated wavetrains, in: AMS Memoirs, vol. 934, Amer. Math. Soc., 2009. [14] E.A. Kuznetsov, P.M. Lushnikov, Nonlinear theory of the excitation of waves by wind due to the Kelvin–Helmholtz instability, JETP 81 (1995) 332–340. [15] T.B. Benjamin, T.J. Bridges, Reappraisal of the Kelvin–Helmholtz problem. Part 2. Interaction of the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instabilities, J. Fluid Mech. 333 (1997) 327–373. [16] T.J. Bridges, Canonical multi-symplectic structure on the total exterior algebra bundle, Proc. R. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006) 1531–1551. [17] T.J. Bridges, Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc. 121 (1997) 147–190. [18] T.J. Bridges, P.E. Hydon, J.K. Lawson, Multisymplectic structures and the variational bicomplex, Math. Proc. Cambridge Philos. Soc. 148 (2010) 159–178. [19] T.J. Bridges, A geometrical formulation of the conservation of wave action and its implications for signature and the classification of instabilities, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 453 (1997) 1365–1395. [20] D.J. Ratliff, T.J. Bridges, Phase dynamics of periodic waves leading to the KP equation in 3 + 1 dimensions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 471 (2015) 20150137. [21] R. Hirota, Exact N-soliton solutions of the wave equation of long waves in shallow water and in nonlinear lattices, J. Math. Phys. 14 (1973) 810–814. [22] G.H. Keulegan, G.W. Patterson, Mathematical theory of irrotational translation waves, J. Res. Natl. Bur. Stand. 24 (1940) 47–101. Available electronically at http://nvlpubs.nist.gov/nistpubs/jres/24/jresv24n1p47_A1b.pdf. [23] F. Ursell, The long-wave paradox in the theory of gravity waves, Math. Proc. Cambridge Philos. Soc. 49 (1953) 685–694. [24] T.J. Bridges, Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow, J. Fluid Mech. 761 (2014) R1. [25] G. Schneider, C.E. Wayne, The long wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math. 53 (2000) 1475–1535. [26] G. Schneider, C.E. Wayne, On the validity of 2D-surface water wave models, GAMM Mitt. Ges. Angew. Math. Mech. 25 (2002) 127–151. [27] M. Chirlius-Bruckner, W.-P. Düll, G. Schneider, Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation, J. Math. Anal. Appl. 414 (2014) 166–175.

The second derivative term in the Boussinesq equation can be included by using an unfolding. Instead of taking Bk = 0 take it to be Bk = ε 2 µ

(10.1)

for some µ = O (1). Then the term Bk qXX is of order ε and appears in the Boussinesq equation modifying (1.13) to 5

Aω qTT + µqXX +



1 2

2



Bkk q + K qXX

=0

(10.2)

XX

with obvious generalization to 2 + 1. The emergence of the two-way Boussinesq equation via modulation can be characterized as ‘‘codimension two Whitham theory’’ in the 1 + 1 case and ‘‘codimension three Whitham theory’’ in the 2 + 1 case as two (1.10), respectively three (1.17), conditions are required for the theory. The codimension can be lowered by the appearance of reflection symmetry. When the governing equations have a transverse reflection symmetry in the y-direction, for example, then the condition Cm = 0 is automatically satisfied when m = 0. Similarly when there is a transverse reflection symmetry in the t-direction then the condition Aω = 0 is automatically satisfied when ω = 0. See §7 of [20] for details of how to show the connection between reflection symmetry and derivatives of (A , B , C ) in the multisymplectic context. Validity of the reduction from a general PDE governed by a Lagrangian to the two-way Boussinesq equation is an open question. However, in special cases, like the reduction from complex Klein–Gordon to two-way Boussinesq in Section 7 there is a potential for a rigorous theory. For example, Chirlius-Bruckner et al. [27] prove validity of the reduction from defocussing NLS to KdV and it appears that similar methods could be applicable to the CKG to Boussinesq reduction. However, determining whether such a theory is possible or carrying out a proof of validity is outside the scope of this paper. Acknowledgments We are grateful to both referees for helpful comments which improved the paper. Special thanks to the referee who pointed us towards the energy argument which is now included in Section 6. The first author is supported by a fully funded EPSRC Ph.D. studentship, and the second author is partially supported by EPSRC grant EP/K008188/1. The second author is grateful to Jerry Bona, Henrik Kalisch, and Guido Schneider for helpful discussions on the

Data statement: Details of the data associated with this paper and how to request access are available from the University of Surrey publications repository: http://epubs.surrey.ac.uk/. References