Conditionally invariant solutions of the rotating shallow water wave equations

# Conditionally invariant solutions of the rotating shallow water wave equations

Benoit Huard
Département de mathématiques et de statistique,
C.P. 6128, Succc. Centre-ville, Montréal, (QC) H3C 3J7, Canada
###### Abstract

This paper is devoted to the extension of the recently proposed conditional symmetry method to first order nonhomogeneous quasilinear systems which are equivalent to homogeneous systems through a locally invertible point transformation. We perform a systematic analysis of the rank- and rank- solutions admitted by the shallow water wave equations in dimensions and construct the corresponding solutions of the rotating shallow water wave equations. These solutions involve in general arbitrary functions depending on Riemann invariants, which allow us to construct new interesting classes of solutions.

## 1 Introduction

In this paper, we use the conditional symmetry method in the context of Riemann invariants (CSM) as presented in  to obtain conditionally invariant solutions of the rotating shallow water wave (RSWW) equations with a flat bottom topography

 ⎧⎪⎨⎪⎩ut+uux+vuy+ghx=2Ωv,Ω∈R,vt+uvx+vvy+ghy=−2Ωu,ht+uhx+vhy+h(ux+vy)=0, (1.1)

where we denote by and the independent and dependent variables respectively. Here, and stand for the velocity vector fields, represents the height of the fluid layer, is the gravitational constant and characterizes the constant angular velocity of the fluid around the -axis induced by a Coriolis force. It can be proved using the chain rule, see , that if a set of functions satisfies the irrotational shallow water wave equations (SWW)

 Δ′(x,u):⎧⎪⎨⎪⎩u′t′+u′u′x′+v′u′y′+gh′x′=0,v′t′+u′v′x′+v′v′y′+gh′y′=0,h′t′+u′h′x′+v′h′y′+h′(u′x′+v′y′)=0, (1.2)

then the functions defined by

 t′=−12Ωcot(Ωt),x′=12(y−xcot(Ωt)),y′=−12(x+ycot(Ωt)),u′=−12(usin(2Ωt)−v(1−cos(2Ωt))−2Ωx),v′=−12(u(1−cos(2Ωt))+vsin(2Ωt)−2Ωy),h′=h2(1−cos(2Ωt)), (1.3)

form a solution of the RSWW equations.

The task of constructing invariant solutions of systems (1.1) and (1.2) using the classical Lie approach was undertaken by several authors. A systematic classification of the subalgebras of the symmetry algebra of the equations describing a rotating shallow water flow in a rigid ellipsoidal bassin was performed in  and many invariant solutions were obtained. In , the author introduced the transformation (1.3) to generate invariant solutions of (1.1) from known invariant solutions of the homogeneous system (1.2), previously computed in  .

The CSM approach to be used in this paper was developed progressively and applied in [4, 9, 8] in order to construct rank-2 and rank-3 solutions to the equations governing the flow of an isentropic fluid. The main feature of this approach, which proved to be less restrictive than the generalized method of characteristics , is that the obtained rank- solutions can depend on many arbitrary functions of many independent variables, called Riemann invariants. Through a judicious selection of these arbitrary functions, it is possible to construct solutions of the considered homogeneous system which are bounded everywhere, even when the Riemann invariants admit a gradient catastrophe . Although the applicability of the CSM approach is technically restricted to first order homogenous hyperbolic quasilinear systems, the objective of the present paper is to apply it to the RSWW equations (1.1) through the transformation (1.3). Large classes of implicit rank- solutions are then constructed for the SWW and RSWW equations, including bumps, kinks and periodic solutions.

The paper is organized as follows. We give in Section 2 the symmetry algebra of system (1.1) and construct the point transformation (1.3) relating systems (1.1) and (1.2). Section 3 contains a brief review of the conditional symmetry method in the context of Riemann invariants for homogeneous systems and we present many interesting rank-1 and rank-2 solutions to the SWW-equations (1.2) together with corresponding solutions to the RSWW equations (1.1). Results and perspectives are summarized in Section 4.

## 2 The symmetry algebra

The classical Lie symmetry algebra admitted by system (1.1) is generated by vector fields of the form

 X=ξ1(x,u)∂t+ξ2(x,u)∂x+ξ3(x,u)∂y+η1(x,u)∂u+η2(x,u)∂v+η3(x,u)∂h. (2.1)

The requirement that the generator (2.1) leave system (1.1) invariant yields an overdetermined system of linear equations for the functions and , . Since this step is completely algorithmic and involves tidy computations, many computer programs have been designed to derive these determining equations, see  for a complete review. The package symmgrp2009.max [1, 11] for the computer algebra system Maxima has been used in this work to obtain the determining equations of the RSWW equations (1.1) and solve them partially in a recursive way. Solving them shows that the Lie algebra of point symmetries of the RSWW equations (1.1) is nine-dimensional and is generated by the following differential generators

 P0=∂t,P1=∂x,P2=∂y,L=y∂x−x∂y+v∂u−u∂v,G1=−12Ωcos(2Ωt)∂x+12Ωsin(2Ωt)∂y+sin(2Ωt)∂u+cos(2Ωt)∂v,G2=12Ωsin(2Ωt)∂x+12Ωcos(2Ωt)∂y+cos(2Ωt)∂u−sin(2Ωt)∂v,D=x∂x+y∂y+u∂u+v∂v+2h∂h,Z1=sin(2Ωt)∂t+Ω[xcos(2Ωt)+ysin(2Ωt)]∂x+Ω[ycos(2Ωt)−xsin(2Ωt)]∂y+Ω[(2Ωy−u)cos(2Ωt)−(2Ωx−v)sin(2Ωt)]∂u−Ω[(2Ωx+v)cos(2Ωt)+(2Ωy+u)sin(2Ωt)]∂v−2Ωhcos(2Ωt)∂h,Z2=cos(2Ωt)∂t+Ω[ycos(2Ωt)−xsin(2Ωt)]∂x−Ω[xcos(2Ωt)+ysin(2Ωt)]∂y−Ω[(2Ωy−u)sin(2Ωt)+(2Ωx−v)cos(2Ωt)]∂u+Ω[(2Ωx+v)sin(2Ωt)−(2Ωy+u)cos(2Ωt)]∂v+2Ωhsin(2Ωt)∂h. (2.2)

The geometrical interpretation of these generators is as follows. The system (1.1) is left invariant by translations in the space of independent variables since it is autonomous. The element generates a rotation of the whole coordinate system while and represent helical rotations. The system is also left invariant by the dilation and the two conformal transformations and .

The Levi decomposition of the symmetry algebra can be exhibited by considering its commutation table (Table 1) in the following basis

 Y1=P2−2ΩG2,Y2=−(P1+2ΩG1),Y3=P1−2ΩG1,Y4=P2+2ΩG2,Y5=−L,Y6=D,Y7=P0−ΩL−Z2,Y8=P0−ΩL+Z2,Y9=−1ΩZ1. (2.3)

Here is a maximal solvable ideal and is isomorphic to the simple Lie algebra . Following the procedure presented in [6, 7], we introduce a set of canonical variables associated with the abelian subalgebra and defined by

 Y7t′=1,Y1t′=0,Y2t′=0,Y7x′=0,Y1x′=1,Y2x′=0,Y7y′=0,Y1y′=0,Y2y′=1,Y7u′=Y7v′=Y7h′=Y1u′=Y1v′=Y1h′=Y2u′=Y2v′=Y2h′=0, (2.4)

to bring system (1.1) into an equivalent autonomous form. It turns out that the set of variables (1.3) satisfies system (2.4) so that when expressed in these variables, the vector fields are rectified to the canonical form

 Y7=∂t′,Y1=∂x′,Y2=∂y′.

Moreover, using the chain rule, it is easily found that system (1.1) transforms to

 u′t′+u′u′x′+v′u′y′+gh′x′=0,v′t′+u′v′x′+v′v′y′+gh′y′=0,h′t′+u′h′x′+v′h′y′+h′(u′x′+v′y′)=0,

which shows the equivalence between systems (1.1) and (1.2). The next section demonstrates how the point transformation (1.3) can be used to construct implicit solutions of equations (1.1) expressed in terms of Riemann invariants.

## 3 Conditionally invariant solutions of the SWW and RSWW equations

We present in this section a brief description of the CSM approach developed progressively in  and  and obtain several rank- and rank- solutions of the SWW equations in closed form. We illustrate the process of construction of the corresponding solutions for the RSWW equations with several interesting examples. The SWW equations (1.2) can be written in matrix evolutionary form as

 ut+a1(u)ux+a2(u)uy=0, (3.1)

where are matrix functions given by

 a1=⎛⎜⎝u0g0u0h0u⎞⎟⎠,a2=⎛⎜⎝v000vg0hv⎞⎟⎠.

The objective is to construct rank- solutions, , of system (3.1) expressible in terms of Riemann invariants. To this end, we look for solutions of (3.1) defined implicitly by the relations

 u=f(r1(x,u),…,rk(x,u)),rA(x,u)=λAi(u)xi,det(λA0I3+a1(u)λA1+a2(u)λA2)=0,A=1,…,k, (3.2)

for some function , where is the by identity matrix. A solution of the form (3.2) will be called a rank- solution if in some open set around the origin, where stands for the Jacobian matrix of in the original variables. The functions are called the Riemann invariants associated with the linearly independent wave vectors , which are obtained by solving the dispersion relation of equation (3.1) for the phase velocity . This relation takes the form

 det(λ0I3+a1(u)λ1+a2(u)λ2)=(λ0+λ1u+λ2v)(λ0+λ1u+λ2v+√gh)(λ0+λ1u+λ2v−√gh)=0. (3.3)

The wave vectors are thus of the entropic (E) and acoustic (S) type defined respectively by

 i)λE=(−λ1u−λ2v,λ1,λ2),ii)λSε=(−(λ1u+λ2v+ε√gh),λ1,λ2),|→λ|2=λ12+λ22=1,ε=±1. (3.4)

We associate to each of them the corresponding Riemann invariant

 i)rE=−(λ1u+λ2v)t+λ1x+λ2y,ii)rSε=−(λ1u+λ2v+ε√gh)t+λ1x+λ2y,|→λ|2=1,ε=±1. (3.5)

The analysis of rank- solutions for the cases are very similar, hence we restrict ourselves to the positive case.

It is convenient when studying solutions of type (3.2) to write system (3.1) in the form of a trace equation,

 Tr[Aμ(u)∂u]=0,μ=1,…,l, (3.6)

where are now matrix functions of , defined by

 A1=⎛⎜⎝100u0gv00⎞⎟⎠,A2=⎛⎜⎝0100u00vg⎞⎟⎠,A3=⎛⎜⎝001h0u0hv⎞⎟⎠.

The construction of rank- solutions through the conditional symmetry method is achieved by considering an overdetermined system, consisting of the original system (3.1) together with a set of compatible first order differential constraints (DCs),

 ξia(u)uαi=0,λAi(u)ξia(u)=0,a=1,…,3−k,A=1,…,k, (3.7)

for which a symmetry criterion is automatically satisfied. Here and throughout this work, we use the summation convention over repeated indices. Introducing the functions

 ¯x1=r1(x,u),…,¯xk=rk(x,u),¯xk+1=xk+1,…¯u=u,¯v=v,¯h=h, (3.8)

as new coordinates on space, the Jacobi matrix now reads

 ∂u=∂f∂r(Ik−(η0t+η1x+η2y)∂f∂r)−1λ, (3.9)

where

 λ=(λAi)∈Rk×3,r=(r1,…,rk)∈Rk,∂f∂r=(∂fα∂rA)∈R3×k, ηa=(∂λAa∂uα)∈Rk×3,a=0,…,2, (3.10)

so that system (3.6) is now expressed as

 Tr[Aμ(u)∂f∂r(Ik−(η0t+η1x+η2y)∂f∂r)−1λ]=0,μ=1,…,l. (3.11)

Requiring that system (3.11) be satisfied for all values of the coordinates , the following result holds (see  for a general statement and a detailed proof).

###### Proposition 1

The nondegenerate quasilinear hyperbolic system of first order PDEs (3.1) admits a -dimensional conditional symmetry algebra , , if and only if there exists a set of linearly independent vector fields

 Xa=ξia(u)∂∂xi,a=1,…,3−k,det(ai(u)λAi)=0,λAiξia=0,A=1,…,k,

which satisfy, on some neighborhood of , the trace conditions

 k=1:i)tr(Aμ∂f∂rλ)=0,μ=1,…,3, (3.12) k=2:i)tr(Aμ∂f∂rλ)=0,ii)tr(Aμ∂f∂rηa∂f∂rλ)=0,a=0,…,2, (3.13)

where the relevant matrices are defined in (3.10). Solutions of the system which are invariant under the Lie algebra are precisely rank- solutions of the form (3.2).

Note that the vector fields , , are not symmetries of the original system. Nevertheless, as we will show, they can be used to build solutions of the overdetermined system composed of (3.1) and the differential constraints (3.7).

To construct solutions of the RSWW equations, we assume that a solution of the SWW equations (1.2)

 u=u(r),v=v(r),h=h(r)r=(r1,…,rk),

has been obtained from equations (3.12) or (3.13). Then the Riemann invariants can be expressed as a graph

 rA=rA(x,u)=rA(x,Φ(r)) (3.14)

in the space for some function . The change of variables (1.3) induces a transformation of the independent variables in this space,

 t→−12Ωcot(Ωt),x→12(y−xcot(Ωt)),y→−12(x+ycot(Ωt)), (3.15)

and we denote by the resulting functions in the new variables. Then, according to transformation (1.3), the functions

 ~u=−u(~r)cot(Ωt)−v(~r)+Ω(y+xcot(Ω)),~v=u(~r)−v(~r)cot(Ωt)−Ω(x−ycot(Ωt)),~h=h(~r)csc2(Ωt), (3.16)

form a solution of the RSWW equations (1.1). Even though tranformation (1.3) is singular at every time , , we show that it is possible to obtain implicit solutions defined in a neigborhood of the origin .

### 3.1 Rank-1 solutions

The reduction procedure outlined above has been applied to obtain rank- and rank- solutions of the SWW equations (1.2) and their corresponding solutions of the RSWW system (1.1). We present here several rank-1 solutions, also called simple waves, associated with the different types of wave vectors (3.4). Note that in the case where , the CSM and the generalized method of characteristics agree .

i) Simple entropic-type waves are obtained by considering system (1.2) in the new variables

 ¯t=t,¯x=r(x,u),¯y=y,¯u=u,¯v=v,¯h=h,

where and the functions , , are allowed to depend on . Following Proposition 1, we look for solutions invariant under the vector fields

 X1=λ1∂t+(λ1u+λ2v)∂x,X2=λ2∂t+(λ1u+λ2v)∂y. (3.17)

The transformed system (3.12) reads as

 gλ1hr=0,gλ2hr=0,(λ1ur+λ2vr)h=0. (3.18)

To obtain a nontrivial solution, we must have together with the relation

 λ1ur+λ2vr=0. (3.19)

For example, if and are constant, we can express in terms of and obtain the explicit solution

 u=u0−λ2λ1v(r),v=v(r),h=h0,r=−u0λ1t+λ1x+λ2y,λ1≠0,h0∈R+,

where is an arbitrary constant and is an arbitrary function.

When the are not constant, different choices can lead to solutions for the velocity vector fields and which are of distinct nature. For example, consider the choice , , leading to

 uur+vvr=12(u2+v2)r=0⇒u2+v2=C2,C∈R.

A periodic solution is obtained by choosing

 u=Csinr,v=Ccosr,h=h0,C∈R, (3.20)

where the Riemann invariant is given implicitly by

 r=−C(Ct−xsinr−ycosr). (3.21)

When , , equation (3.19) implies

 vur+uvr=(uv)r=0⇒v=Cu(r),C∈R.

We then get the solution

 u=u(r),v=Cu(r),h=h0∈R,r=−2Ct+Cu(r)x+u(r)y, (3.22)

where is an arbitrary function of the Riemann invariant .

ii) Similarly, simple acoustic-type waves are obtained by considering system (1.2) in the new variables

 ¯t=t,¯x=r(x,u),¯y=y,¯u=u,¯v=v,¯h=h,

where , , and the functions , , are allowed to depend on . Rank-1 solutions of this type are invariant under the vector fields

 X1=λ1∂t+(λ1u+λ2v+√gh)∂x,X2=λ2∂t+(λ1u+λ2v+√gh)∂y. (3.23)

In this case, the transformed system (3.12) is

 λ1√ghhr=ur,λ2√ghhr=vr,h(λ1ur+λ2vr)=√ghhr. (3.24)

The third equation is automatically satisfied whenever the first two are and . Note that in order to obtain a solution for , it is necessary that the relation

 λ1(u,v,h)vr−λ2(u,v,h)ur=0 (3.25)

be satisfied. Considering different choices for the functions , we obtain several interesting solutions, presented in Table 2.

For illustration, we now turn to the construction of the implicit solution of the RSWW equations corresponding to (3.20), (3.21) using transformation (1.3). We first transform the Riemann invariant to obtain an implicit equation for ,

 (3.26)

Using equations (3.16), we obtain the implicit solution of the RSWW equations

 u=−Ccos~r−Ccot(Ωt)sin~r+Ω(y+xcot(Ωt)),v=Csin~r−Ccot(Ωt)cos~r−Ω(y+xcot(Ωt)),h=h0csc2(Ωt), (3.27)

where is the solution of the implicit equation (3.26). This solution has period and goes to infinity at every time , . Nevertheless, due to the invariance of equations (1.1) with respect to translations in time, it is possible to use a time shift so that equations (3.26) are well defined in a neighborhood of length around . For example, the translation gives the solution

 u=−Ccos¯r+Ctan(Ωt)sin¯r+Ω(y−xtan(Ωt)),v=Csin¯r+Ctan(Ωt)cos¯r−Ω(y−xtan(Ωt)),h=h0sec2(Ωt), (3.28)

where satisfies the equation

 ¯r=−C22Ωtan(Ωt)+C2[(y+xtan(Ωt))sin¯r−(x−ytan(Ωt))cos¯r], (3.29)

which is clearly defined in the interval . Note that this process can be applied to every solution presented in Table 2 to generate local solutions of the RSWW equations defined around .

### 3.2 Rank-2 solutions

The construction of rank- solutions is much more involved than in the case since it requires us to solve system (3.13), which is composed of at most twelve independent nonlinear partial differential equations, compared to only three equations. However, we now show that the task is undertakable and leads to interesting solutions. The results of this analysis are summarized in Table 3 and 4.

i) We first look for rank- solutions resulting from the interaction of two entropic-type solutions. They are invariant under the vector field

 X=∂t+u∂x+v∂y. (3.30)

In the variables

 ¯t=t,¯x1=r1(x,u),¯x2=r2(x,u),¯u=u,¯v=v,¯h=h,ri(x,u)=t−λi1λi1u+λi2vx−λi2λi1u+λi2vy,i=1,2, (3.31)

 λ11(λ21u+λ22v)hr1+λ21(λ11u+λ12v)hr2=0, (3.32) λ12(λ21u+λ22v)hr1+λ22(λ11u+λ12v)hr2=0, (3.33) (λ21u+λ22v)(λ11ur1+λ12vr1)+(λ11u+λ12v)(λ21ur2+λ22vr2)=0. (3.34)

A solution to the first two equations exists if and only if

 (λ11λ22−λ12λ21)(λ11u+λ12v)(λ21u+λ22v)=0orh=h0∈R+.

The conditions on the functions imply either that the wave vectors are parallel or one of the considered waves has zero velocity. From these conditions, we now show that no rank- solution can be built from this type of interaction.

When , the Riemann invariants and are equal, hence the solution cannot be of rank . Therefore we look for solutions with , a positive constant. Equation (3.34) implies that

 ur1=−1λ11λ11u+λ12vλ21u+λ22v(λ21ur2+λ22vr2)−λ12λ11vr1. (3.35)

We then consider the linear combination

 1uvTr[A3∂f∂r(uη1+vη2)∂f∂rλ]=−2uv(λ11u+λ12v)(λ21u+λ22v)(λ11λ22−λ12λ21)×((λ11u+λ12v)(λ21ur2+λ22vr2)vr2+(λ21u+λ22v)(λ11ur2+λ12vr2)vr1), (3.36)

implying that a rank-2 solution must satisfy

 (λ11u+λ12v)(λ21ur2+λ22vr2)vr2+(λ21u+λ22v)(λ11ur2+λ12vr2)vr1=0. (3.37)

When , equation (3.35) requires that

 ur1=−λ12λ11vr1,vr2=−λ21λ22ur2,

so that (3.37) becomes

 1λ22(λ21u+λ22v)(λ11λ22−λ21λ12)ur2vr1=0,

leading necessarily to a rank- solution. Hence we can solve (3.37) for , and the expression (3.35) for implies that

 ur2ur