^{1,2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

Generalized solutions of the shallow water equations are obtained. One studies the particular case of a generalized soliton function passing by a variable bottom. We consider a case of discontinuity in bottom depth. We assume that the surface elevation is given by a step soliton which is defined using generalized solutions (Colombeau 1993). Finally, a system of functional equations is obtained where the amplitudes and celerity of wave are the unknown parameters. Numerical results are presented showing that the generalized solution produces good results having physical sense.

The classical nonlinear shallow water equations were derived in [

Shallow water equations have been submitted to numerous improvements to include several physical effects. In such sense, several dispersive extensions were developed. The inclusion of dispersive effects resulted in a big family of the so-called Boussinesq-type equations [

However, there are a few studies which attempt to include the discontinuous or not differentiable bottom effect into shallow water equations [

In this paper, we relaxedly completed this hypothesis allowing that the bottom function must be not differentiable by using the Colombeau algebra [

The method presented in this paper is general, and it can be used for a wide class of nonlinear dispersive wave equations such as Boussinesq-like system of equations. In order to try the possibilities of this theory, we consider the equation deduced in [

To study the nonlinear and bottom irregularities effects, we consider the shallow water equations to simulate a generalized soliton passing by discontinuity in the bottom. The idea of taking a soliton to describe a traveling wave and singular solution as a soliton was developed by several works [

The starting point, that the bottom has a discontinuity, constitutes a generalization of submerged structure or coral reef representation. This situation is equivalent, in practical engineering, to the presence of a vertical hard structure that in some cases breaks the wave propagation. As a wave propagates over the structure, part of the wave energy is reflected back to the open ocean, part of the energy is transmitted to the coast, and part of the energy is converted to turbulence and further dissipated in the vicinity of the structures [

In this paper, we obtain generalized solutions of the shallow water equations in the one-dimensional case. The approximate solution is obtained as a singular solution. We suppose that in microscopic sense, when a wave crosses the discontinuity bottom point, one part continues its propagation to the shore, preserving the initial structure, while another part is reflected. We use a generalized soliton function which has macroscopic aspect in sense of Colombeau [

This paper begins with a description of the Colombeau algebra. Some useful proposition including different product of generalized function was established to simplify some nonlinear operations. After that, generalized solutions are obtained for the flat bottom for two types of shallow water equations. In both cases the generalized solution is compared with previous formulas. Finally, we propose a method to obtain the generalized solution in the discontinuous bottom case. The accuracy of the numerical scheme for solving the shallow water equations was verified by comparing the numerical results with the theoretical solutions obtained by [

In this paper, we use a generalized solution deduced from the algebra of Colombeau [

The mathematical theory of generalized solutions allows to obtain new formulas and numerical results [

The simplified algebra of generalized functions is the quotient space

The elements

Two generalized functions

In the interpretation of the generalized solution, we use that two different generalized functions associated with the same distribution differ by an infinitesimal.

It is well known from the classical asymptotical method that the several solutions depend on an infinitesimal

The generalized functions have useful properties for our purpose:

let

it is possible to define the integral of generalized functions in the following way: let

The association

A generalized function

The Heaviside generalized functions are associated between them. Moreover,

A generalized function

It is possible to check that the relation

For a given

For instance,

Sketch of a representative of step soliton generalized function.

From Definition

A generalized function

From Definition

Reviewing cases of the product of two step generalized functions, the product with function Heaviside generalized function, and the product derivatives of step generalized functions, as well as products with the microscopic generalized functions, it should be noted that the depth with a discontinuity is closer to the combination of the Heaviside generalized functions. In the calculations with generalized function on the shallow water equations arise the derivatives of Heaviside generalized functions which are reasonably approximated by delta generalized function. In short, in the upcoming paragraph, we show those useful lemmas of the product of generalized functions that allow to simplify the calculations and obtain in this way algebraic equations.

To prove the main results of this paper these lemmas of generalized functions are needed. Such lemmas consist in simplifing association between the product of several generalized functions that appears in the algebras of substitution of the proposal solution in the shallow water equations. Let us prove the following.

Given

We have that

It possible to check that for

Given

We have that

The following propositions are useful.

Given

We prove here that (ii) the others are similar. We have that

Given

We consider the so-called shallow water equations in one dimension as given in [

Schematic diagram of a solitary wave propagating over a mild slope bottom.

Particles in a vertical plane at any instant always remain in a vertical plane, that is, the streamwise velocity is uniform over the vertical. Each vertical plane always contains the same particles; hence, the integration volume is moving with the fluid.

With the previous assumption, we have chosen a material reference frame to describe the motion of the soliton in the fluid.

For a given

It is assumed that solitons of the system (

Using that

The choice of the particle velocity

Taking off the amplitude wave

Theorem

Let us denote

Quotient of wave celerity for two formulations. In the case (a), only nonlinear effect was simulated. In the case (b), the dispersive and nonlinear have the same order

Nonlinear effect

Nonlinear and dispersive of the same order

Also, when

We consider the following so-called shallow water equations with dispersive effect in one dimension as given in [

The following theorem holds.

It is assumed that solitons of the system (

Since the proof is similar to Theorem

Taking

From (

Comparison of wave celerity for dispersive and nonlinear of the same order

In this section, we studied the case in which a soliton crosses a bottom discontinuity (see Figure

Schematic diagram of a solitary wave propagating over a discontinuity bottom.

Following the same idea as in the previous section, we obtain a generalized solution of shallow water equation stated in [

Given

It is assumed that solitons of the system (

Substituting (

Theorem

Now, we obtain a solution of shallow water equation as two solitons which we assume are the propagate soliton, and reflected by the jump. Using the heuristic considerations despite in Remark

Given

For given

Denote that by

Although Theorem

In Theorem

In this section, we show a numerical procedure to find the unknown parameters

Let it be assumed that a generalized solutions of (

Equation (

Let us denote

Taking

In this section, we show that the generalized solutions with physical sense can be obtained. To do so, the constant

The initial values of quasi-Newton method for solving (

In [

We take the example described in [

Theoretical and predicted amplitude wave of the soliton propagating over a discontinued bed. The point

In [

Although we have been adjusted the method well to both theoretic and experimental data, this result constitutes a first approximation of application of Colombeau’s algebra, because we consider as a constant in time and space the amplitude of step soliton generalized function. Also, we do not consider here the friction effect and the time dependency amplitude wave. An other facility is that the parameter

In this paper generalized solutions in the sense of Colombeau of Shallow water equations are obtained. This solution is consistent with numerical and theoretical results of a soliton passing over a flat or discontinuity bottom geometries. The method developed in this paper reduces the partial differential equation to determine the zeros of a functional equation. This procedure also will allow us to study a propagation of several types of singularities on several bottom geometries.

The authors are grateful to Herminia Serrano Mendez for their collaboration. They thank the oceanographist Alina Rita Gutierrez Delgado for helpful discussions and review of the paper. They are also very pleased of the reviewers who helped them improve the paper. The authors appreciate the help of Dan Marchesin and special thanks for Iucinara Braga. They also thank IMPA, Brazil and the University of Université des Antilles et de la Guyane.