Symmetry Properties and Solutions of Shallow Water Equations

Within one-dimensional model of shallow water it is investigated a one-parameter family of equations describes the propagation of surface waves above a straight bottom. The parameter of the family is the slope of the bottom. This family is generated by the equations of one-dimensional model of shallow water with a horizontal bottom. By means of the method of A operators it is found that this system has an infinite aggregate of non-trivial zero-order conservation laws generated by the system of linear differential equations. As a result of special choice of the hodograph transformation, the system of equations of one-dimensional model of shallow water with a horizontal bottom is generated by the same system of linear differential equations. The group analysis of the systems is carried out. An infinite aggregate of non-degenerate solutions of the equations of one-dimensional model of shallow water with a straight bottom is received. All of the degenerate solutions of these equations are found. Thus, the data base of exact solutions of the equations of one-dimensional model of shallow water with a straight bottom is created. The solutions obtained in this paper may be used in the study of tsunami waves and fluid distribution in channels.


Conservation Laws for the System (3)
The conservation law of zero-order for the system (3) is called [8,9] the type correlation made because of the equations of the system are the total differentiation operators in the variables x t, respectively.
The two equations of the system (3) are the conservation laws of zero-order. To find all nontrivial conservation laws of zero-order, is used the method of A -operators proposed by Yu. A. Chirkunov [9][10][11][12][13]. The conservation law is taken as a generating one. It is defined by the second equation of the system (3).
From the system of constitutive equations obtained from the correlation is an evolutionary operator of generalized symmetry [8][9][10][11][12][13] follows, that the factor set of zero-order A -operators for the system (3) by the range of its trivial A -operators is generated by the operators: where function is a solution of an equation and function The operator rule (6) for the components of the conservation law assumed by the formulae (5), generates the components 0 A , 1 A of all nontrivial conservation laws of zero-order for the system (3). Precisely: After the substitution the equation (7) The condition of consistency of correlations (10) generates the equation Thus, the system (3) has an infinite aggregate of non-trivial zero-order conservation laws determined by the correlation (4), where  (12)

Conservation Laws for the System (1).
Let 0 ≠ k . Direct calculations determine that for any pair of functions are the components of an arbitrary zero-order conservation law (4) for the system (3), then the functions comply with the correlation due to the system equations (1) i.e. they are the components of the zero-order conservation law for the system (1).
are the components of an arbitrary zero-order conservation law (14) for the system (1) where 0 ≠ k , then the functions are the components of the zero-order conservation law (4) for the system (3).
then the functions are the components of the zero-order conservation law (4) for the system (3).
In this way, the components of all zero-order conservation laws (14) for the system (1) are determined by the formulae (11)-(13).

The Symmetry Properties of the System (3)
The operator admitted by the system (3) is sought in the form The condition of invariance with respect to this operator of manifold, determined by the system (3) after the splittance according to the parameter derivatives, generates the system of constitutive equations which ends in involution and integrates. As a result of it, the system (3) admits a pseudogroup of Lie transformations generated by the operators where is any solution of the system (12). Thus, the set of zero-order conservation laws for the system (3) and its main group of Lie transformations are determined by the same system of linear equations (12).
On the strength of Lie theorem [9], the formulae of solution production ("birth") are generated from the symmetry property (14). If is the solution of the system (3), then the solution of the system (3) written for the is the following pairs of functions 3  2  1  4  3  1   2  3   5  3  2  1  4  3  1  2  3   ,   ,  ,   a  t  a  a  x  a  a  t  a  a  h  a  h   a  t  a  a  x  a  a  t  a  a  u  a  a  u , is any solution of the system (12).
The formulae of solution production of the system (1) where 0 ≠ k are generated on the strength of the transformation is the solution of the system (1), then the following pairs of functions are the solution of the system (1) written for the variables

Non-degenerate Solutions
The solution ( ) of the system (3) will be called a non-degenerate one in case it is different from (1)  14 Symmetry Properties and Solutions of Shallow Water Equations By the set of non-degenerate solutions the system (3) becomes linear with the help of hodograph transformation, which is convenient to choose in accordance with the conservation laws (11) of the system (3) and its symmetry property (15): As the result of this transformation, the system (3) is equated with the linear system (12). The solutions of the system (3) and (12) are linked in the following way. Any non-degenerate solution , , , , of the system (3) determines a non-degenerate solution of the system (12) by the formulae (23), i.e. the solution . On the strength of (23) and (12), it means that ( ) 0 any solution of the system (12) with const ≠ Φ is a non-degenerate one.
All of the degenerate solutions of the system are as follows where c is an arbitrary real constant.
Conversely, any non-degenerate solution (12)  Thus, the system of linear differential equations (12) determines for the system of equations (3) of one-dimensional model of shallow water with a horizontal bottom, the range of zero-order conservation laws, its main group of Lie transformations, and its set of non-degenerate solutions. On the strength of the transformation (2), it is also true for the system (1) of the equations of one-dimensional model of shallow water with a sloping bottom. Next, the group analysis of the system (12) is carried out. The system's invariant and partially invariant solutions make it possible to generate zero-order conservation laws for the systems (1), (3) of the equations of one-dimensional model of shallow water with a straight bottom and their non-degenerate exact solutions.

The Symmetry Properties of the System (12)
The operator admitted by the system (12) is sought in the form The condition of invariance with respect to this operator of manifold, determined by the equations (12) after the splittance according to the parameter derivatives, generates the system of constitutive equations which ends in involution and integrates after the third extension. As a result of it, the main group of Lie transformations of the system (12) (factor group by normal subgroup linked with the system linearity) is generated by the operators ( ) On the strength of Lie theorem [9] and the system linearity (12), the following formulae of its solution production are generated by the symmetry property (25). It is relevant for any pair of complex-valued functions which are the solution of the system (12). The solution of the system (12) are arbitrary real constants.
On the strength of the infinitesimal formula of solution production [9], the symmetry property (25) and the system linearity (12) generate the following formulae of its solution production. Precisely, if is an arbitrary smooth solution of the system (12) then the solutions of the system are the following pairs of functions Owing to the system linearity (12), any linear combination of solutions of this system is its solution.

Exact Solutions of the System (12)
To classify the invariant and partially invariant solutions of the system (12), the optimal system of dissimilar Lie subalgebras with the basis (25) is set up. Operator 4 X is its center. Each subalgebra of this optimal system of subalgebras corresponds with a subgroup of the main group of the system (12) generated by it. As a result, we have an optimal system of subgroups of the main group of the system. It contains four one-parameter subgroups 4 X , The substituting in the system (12) yields the factor system , which reduces to Hein equation [14].
The substituting in the system (12) yields the factor system ( ) ( ) 0 The substituting in the system (12) reduces to the linear algebraic system of equations relative to h u Ψ Ψ , Similarly to the previous solution it is stated that the equality ( ) λ λ ± = f is a necessary condition for irreducibility [9] of this partially invariant solution to an invariant solution.
The solution of the system (46) where ( ) where c is an arbitrary real constant and f is a nonconstant continuously differentiable function. The solutions (58), (59) are given in [7].

Degenerate solutions of the system (1)
where 0 ≠ k .
i.e. a non-trivial degenerate solution of the system (1) is its partially invariant solution of 1 rank and defect 1.
The substituting (60) in (1) generates The consistency condition of the system (61) is an equality from which it follows that function F is as follows