Conditionally invariant solutions of the rotating shallow water wave equations
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.
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
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)
then the functions defined by
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
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
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
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
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
Moreover, using the chain rule, it is easily found that system (1.1) transforms to
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
where are matrix functions given by
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
The wave vectors are thus of the entropic (E) and acoustic (S) type defined respectively by
We associate to each of them the corresponding Riemann invariant
The analysis of rank- solutions for the cases are very similar, hence we restrict ourselves to the positive case.
where are now matrix functions of , defined by
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),
for which a symmetry criterion is automatically satisfied. Here and throughout this work, we use the summation convention over repeated indices. Introducing the functions
as new coordinates on space, the Jacobi matrix now reads
so that system (3.6) is now expressed as
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
which satisfy, on some neighborhood of , the trace conditions
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)
in the space for some function . The change of variables (1.3) induces a transformation of the independent variables in this space,
and we denote by the resulting functions in the new variables. Then, according to transformation (1.3), the functions
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
where and the functions , , are allowed to depend on . Following Proposition 1, we look for solutions invariant under the vector fields
The transformed system (3.12) reads as
To obtain a nontrivial solution, we must have together with the relation
For example, if and are constant, we can express in terms of and obtain the explicit solution
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
A periodic solution is obtained by choosing
where the Riemann invariant is given implicitly by
When , , equation (3.19) implies
We then get the solution
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
where , , and the functions , , are allowed to depend on . Rank-1 solutions of this type are invariant under the vector fields
In this case, the transformed system (3.12) is
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
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 ,
Using equations (3.16), we obtain the implicit solution of the RSWW equations
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
where satisfies the equation
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
In the variables
equations (3.13 i) read as
A solution to the first two equations exists if and only if
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
We then consider the linear combination
implying that a rank-2 solution must satisfy
When , equation (3.35) requires that
so that (3.37) becomes