Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 910659, 24 pages
http://dx.doi.org/10.1155/2012/910659
Research Article

Step Soliton Generalized Solutions of the Shallow Water Equations

1Department of Fluid Dynamic, IMPA, Dona Castorina 110, Jardín Botànico, 22460-320 Rio de Janeiro, RJ, Brazil
2Oceanology Institute, Environmental Agency, Avenida Primera, 18406, Flores, Playa, 11600 C. Habana, Cuba
3Laboratoire AOC, Université des Antilles et de la Guyane, Campus de Fouillole, 97159 Pointe-à-Pitre, Guadeloupe
4Departamento de Matemáticas, Facultad de Matemáticas y Computación, Universidad de la Habana, San Lázaro esq. A L, 10400 La Habana, Cuba

Received 5 January 2012; Revised 11 April 2012; Accepted 19 April 2012

Academic Editor: Armin Troesch

Copyright © 2012 A. C. Alvarez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

The classical nonlinear shallow water equations were derived in [1]. There exist several works devoted to the applications, validations, or numerical solutions of these equations [25]. These equations provide a significant improvement over linear wave theory to describe the wave-breaking process [6].

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 [710]. Many other families of dispersive wave equations have been proposed as well [1113]. Other studies attempt to include the effect of different types of bottom shape [4, 1419]. Also in [20, 21], the mild slope hypothesis is not required, and rapidly varying topographies was also considered. In these studies, the asymptotical expansion method was used. In [22] was included different geometry bathymetric by improving the shallow water equations by using variational principles.

However, there are a few studies which attempt to include the discontinuous or not differentiable bottom effect into shallow water equations [2325]. One reason is that in its deduction procedure, assume certain restrictions on bottom type function as differentiability. In [3], a numerical method to studding the discontinuous bottom was used.

In this paper, we relaxedly completed this hypothesis allowing that the bottom function must be not differentiable by using the Colombeau algebra [26, 27] studying the shallow water equations with a discontinuous bottom. This algebra comes being used in several applications of the physics fields studying nonlinear partial differential equation. In this theory, the previous solutions are still valid because of the natural embedding of the distribution in the sense of Schwartz in this algebra. In particular the smooth functions are embedded as a constant sequence. However, this theory is specially useful when the product distribution is not allowed or when a formalism of continuous function is not more valid. Details of Colombeau algebra in the applications to hydrodynamics of can be found in [28].

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 [6] with the principles that the equality 𝑝𝑝𝑜=𝜌𝑔(𝑜+𝜂) holds; here, 𝑝 is the pressure, 𝑜 is the depth, 𝜌 is the density of the water, and 𝜂 is the surface elevation. This equality for the discontinuous bottom case is not more valid in the classical sense. So, we embedded the classical distribution in the generalized function where the nonlinear operations are allowed. Also, we consider the dispersive equations deduced in [8] which is valid to variable smooth bottom. Similar formulas obtained in this paper were obtained in [29] by using the method of the lines.

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 [3035]. In [36], a generalized solution in the frame of Colombeau’s generalized functions was obtained. Solitons are used in coastal engineering to describe waves approximating to the coast with the presence of a vertical structure [3741]. The evolution of a solitary wave at an abrupt junction was measured and discussed by [42] in detail. There exist a number of physical reasons to suppose that the propagation of a soliton wave over a discontinuity point in the bottom preserves the shape and the structure of an initial wave [43].

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 [39, 44]. These processes we approximated by using two generalized solitons traveling in opposite direction.

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 [27], that is, for a given 𝜏1, 𝑆𝜏1(𝜏)=0 for 𝜏<𝜏1, 𝑆𝜏1(𝜏)=1 for 𝜏1<𝜏<𝜏1 and 𝑆𝜏1(𝜏)=0 for 𝜏>𝜏1. We obtain a nice procedure that reduces the problem of finding a solution of nonlinear partial differential equation to the one of solving a system of algebraic equations. Since this attempt used this theory to obtain practical formulas, we prove that in the limit, the Step generalized solution agreement reasonably with previous classical solutions. Moreover, we prove that by fixing some parameter that appears in this theory, some nonlinear and dispersive effects are reproduced well.

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 [45] and experimental data obtained in [46].

2. Colombeau Algebra

In this paper, we use a generalized solution deduced from the algebra of Colombeau [27, 47]. Such solution permits to construct a singular solution of the system of conservation law that preserves its structures and initial shape. These functions appear in the multiplication of distributions theory when nonlinear differential equations are studied.

The mathematical theory of generalized solutions allows to obtain new formulas and numerical results [48]. The method proposed in [26, 49] is quite general, but each particular problem requires the definition of specific generalized functions. A general definition can be found in the specialized literature (see as an example [26, 27, 47]). Here, we present a version which is sufficient for the purpose of this paper. Let Ω be an open subset in . Putting 𝐸𝑠𝑅(Ω)=1(𝜖,𝑥)(0,1)×Ωsuchthat𝑅1𝐶,𝐸(Ω),𝜖(0,1)𝑀𝑅(Ω)=1𝐸𝑠||(Ω)/compact𝐾Ωandforalldierentialoperator𝐷𝑞,𝑐>0,𝜂>0suchthat𝐷𝑅1||(𝜖,𝑥)𝑐𝜖𝑞,𝑁𝑅,𝑥𝐾,0<𝜖<𝜂(Ω)=1𝐸𝑀||(Ω)/(𝐾)Ωcompactand𝐷dierentialoperator,𝑞,𝑝𝑞,𝑐>0,𝜂>0,suchthat𝐷𝑅1||(𝜖,𝑥)𝑐𝜖𝑝𝑞,,𝑥𝐾,0<𝜖<𝜂(2.1)

𝐸𝑠(Ω) and 𝐸𝑀(Ω) are algebras, and 𝑁(Ω) is an ideal of 𝐸𝑠(Ω).

Definition 2.1. The simplified algebra of generalized functions is the quotient space 𝑠(Ω)=𝐸𝑠(Ω)/𝑁(Ω).

The elements 𝐺 of 𝑠(Ω) are denoted by 𝐺=𝑅1(𝜖,𝑥)+𝑁(Ω). Distribution of compact support on can be embedded on 𝑠() by convolution with a mollifier 𝜌𝜖, defined as follows: let 𝜌𝑆() (Schwartz’s space) with the properties 𝜌(𝑥)𝑑𝑥=1, 𝑥𝛼𝜌(𝑥)𝑑𝑥=0, for all 𝛼𝑁2,|𝛼|>1, then we set 𝜌𝜖(𝑥)=(1/𝜖)2𝜌(𝑥/𝜖). Then the generalized function 𝑥𝛼𝜌𝜀(𝑥𝑦)𝑤(𝑥)𝑑𝑥+𝑁(Ω) belongs to 𝑠() [28].

𝑠(Ω) is clearly an algebra with the usual pointwise operations of addition, inner multiplication, and exterior multiplication by scalars. In this algebra, there are two equalities, one strong (=) and one weak (). The strong one is the classical algebraic equality. The weak one is called association and is denoted by the symbol ; in other words, two simplified generalized functions are equal if the difference of two of their representatives belongs to the ideal 𝑁(Ω). Also, whereas multiplication is compatible with equality in 𝑠(Ω), it is not compatible with association. Therefore, the distinction between (=) and () automatically ensures that the physically correct solution is selected, a distinction that can be made in analytical as well as in numerical calculations by using a suitable algorithm [27, 28].

Definition 2.2. Two generalized functions 𝐺1,𝐺2𝑠(Ω) are associated, 𝐺1𝐺2, if there exists representatives 𝑅1,𝑅2𝐸𝑠(Ω) of 𝐺1, 𝐺2 respectively, such that: for all 𝜓𝐷(),   (𝑅1(𝜖,𝑥)𝑅2(𝜖,𝑥))𝜓(𝑥)𝑑𝑥0when𝜖0.

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 𝜖. For example, in [6, page 470], the solution of the Korteweg de Vries is given in linear limit as 𝜍𝜖=𝜆cos(𝜖(𝑥𝑐𝑡)), whereas in the solitary waves limit as 𝜍𝜖=𝜆sech(𝜖(𝑥𝑐𝑡)). In [36], similar solutions are obtained in the sense of Colombeau. These functions show that even in the classical sense, the solution is given by a family of functions. The idea to look for a generalized solution in the sense of Colombeau means to seek a solution like a family that depend of one infinitesimal, but this extension must guarantee that they keep valid the association by differentiation and nonlinear operations between them.

The generalized functions have useful properties for our purpose:(i)𝐶(Ω)𝑠(Ω), (ii)let 𝜌𝐷() be a 𝐶() function such that 𝜌(𝑥)𝑑𝑥=1. Then the class of 𝑅1(𝜖,𝑥)=(1/𝜖)𝜌(𝑥/𝜖) is an element of 𝑠(Ω) associated with the Dirac delta function, that is, for all 𝜓𝐷(), 𝑅1(𝜖,𝑥)𝜓(𝑥)𝑑𝑥𝜓(0), when 𝜖0, where 𝐷() denotes the space of the infinitely smooth functions on with compact support.(iii)it is possible to define the integral of generalized functions in the following way: let 𝐺𝑠() and 𝑅1𝐸𝑠() a representative. The application 𝑅2(𝜖,𝑥)(0,1)× is defined by (𝜖,𝑥)𝑅2(𝜖,𝑥)=𝑥𝑥𝑜𝑅1(𝜖,𝑥)𝑑𝑥,(2.2)then 𝑅2𝐸𝑠(), for all 𝑥𝑜. The class 𝐽𝑠() of 𝑅2 verifies 𝑑𝐽/𝑑𝑥=𝐽=𝐺 and is called a primitive of 𝐺.

The association is stable by differentiation but not by multiplication, that is, if 𝐺1,𝐺2,𝐺𝑠(), and 𝐺1𝐺2 then 𝐺1𝐺2, but 𝐺𝐺1 and 𝐺𝐺2 are not necessarily associated.

Definition 2.3. A generalized function 𝐻𝑠() is called a Heaviside generalized function if it has representative 𝑅𝐸𝑠() such that there exists a sequence of real numbers 𝐴(𝜖)>0, 𝐴(𝜖)0, when 𝜖0 such that(i)𝑅(𝜖,𝑥)=0, for all 𝜖>0, and 𝑥<𝐴(𝜖),(ii)𝑅(𝜖,𝑥)=1, for all 𝜖>0, and 𝑥>𝐴(𝜖),(iii)sup|𝑅(𝜖,𝑥)|<+, 𝜖>0, and 𝑥.

The Heaviside generalized functions are associated between them. Moreover, 𝐻𝑛𝐻 for 𝑛, 𝑛>0.

Definition 2.4. A generalized function 𝛿𝑠() is called Dirac generalized function if it has a representative 𝑅𝐸𝑠() such that there exists a sequence 𝐴(𝜖)>0, 𝐴(𝜖)0, when 𝜖0 such that(i)𝑅(𝜖,𝑥)=0, for all 𝜖>0, and |𝑥|>𝐴(𝜖),(ii)𝑅(𝜖,𝑥)𝑑𝑥=1, for all 𝜖>0,(iii)|𝑅(𝜖,𝑥)|𝑑𝑥<C, for all 𝜖>0, where C is a constant independent of 𝜖.

It is possible to check that the relation 𝐻𝛿 holds between Heaviside and Dirac generalized functions. Moreover, for a reasonable Heaviside and Dirac generalized function, there exists a constant 𝑀 such that 𝐻𝛿𝑀𝛿.

Definition 2.5. For a given 𝜏1>0, a generalized function 𝑆𝜏1𝑠() is called a step soliton generalized function if it has a representative 𝑅𝐸𝑠() defined by(i)𝑅(𝜖,𝑥)=𝑅1(𝜖,𝑥𝜏1)𝑅2(𝜖,𝑥+𝜏1),where 𝑅1,𝑅2𝐸𝑠() are representative, of a Heaviside generalized function.

For instance, 𝑅1(𝜖,𝑥)=0 if 𝑥+𝜏10, 𝑅1(𝜖,x)=1 if 𝑥+𝜏1𝜀, and 𝑅1(𝜖,𝑥)>0 if 0<𝑥+𝜏1𝜀 (see Figure 1(a)). Besides, 𝑅1(𝜖,𝑥)=0 if 𝑥𝜏10, 𝑅1(𝜖,𝑥)=1 if 𝑥𝜏1𝜀, and 𝑅1(𝜖,𝑥)>0 if 0<𝑥𝜏1𝜀 (see Figure 1(b)). In Figure 1(c), the graph of 𝑅1(𝜖,𝑥+𝜏1)𝑅1(𝜖,𝑥𝜏1) is shown.

fig1
Figure 1: Sketch of a representative of step soliton generalized function.

From Definition 2.5, we obtain that the equality 𝑆𝜏1(𝑥)=𝐻(𝑥+𝜏1)𝐻(𝑥𝜏1) holds. Moreover, the macroscopic aspect of the step generalized function is not necessarily symmetric (see Figure 1). A lesson from this application is that by assuming that physically relevant distributions such as Heaviside 𝐻 and Dirac 𝛿 generalized function are elements of 𝑠(); one gets a picture that is much closer to reality than if they are restricted to classical sense. This fact can be exploited in mathematical and physical modeling. We can verify that the step generalized soliton has one as the maximum value of its representatives. Thus, it is possible to verify that the generalized function 𝜆𝑆𝜏1 has 𝜆 as the maximum values.

Definition 2.6. A generalized function 𝛿1𝑠() is called a microscopic soliton generalized function if it has a representative 𝑅𝐸𝑠() defined by(i)𝑅(𝜖,𝑥)=1𝑅1(𝜖,𝑥)𝑅1(𝜖,𝑥),where 𝑅1𝐸𝑠() is a representative of a Heaviside generalized function.

From Definition 2.6, we obtain that the relation 𝛿1(𝜏)=(1𝐻(𝜏)𝐻(𝜏)) holds. Moreover, 𝛿1 generates a family of generalized functions with different height 𝛾, that is, 𝛿𝛾(𝜏)=𝛾𝛿1(𝜏). Let us denote by 𝜃 the function that satisfies 𝜋𝜃(𝑥)=0,for𝑥<0,𝜃(𝑥)=2,for𝑥=0,𝜃(𝑥)=0,for𝑥>0.(2.3) Then the function 𝜃 has the macroscopic aspect of the generalized function 𝛿𝜋/2=(𝜋/2)𝛿1. Then we have 𝛿𝜋/2=𝜃,(2.4) where 𝜃 is given in (2.3), and 𝛿𝜋/2 is the microscopic soliton with height 𝜋/2. Let us define the composite function 𝜋cos(𝜃(𝑥))=0,for𝑥<0,cos(𝜃(𝑥))=2,for𝑥=0,cos(𝜃(𝑥))=0,for𝑥>0,(2.5) where 𝜃(𝑥) is given in (2.3). It is possible to check that the generalized function cos(𝜃(𝑥)) has the macroscopic aspect of the generalized function 1𝛿1, where 𝛿1 is the microscopic soliton of height one, that is, 𝛿cos𝜋/2=1𝛿1.(2.6) Let us denote 𝜗(𝑥)=𝜗1,for𝑥<0,𝜗(𝑥)=𝜗2,for𝑥>0,(2.7) with real numbers 𝜗1>𝜗2. Now, using the Heaviside generalized function 𝐻, we can write 𝜗(𝑥)=𝜗1+𝜗2𝜗1𝐻(𝑥).(2.8) We can check that the angle 𝜃 in respect of axis 𝑂𝑋 in each point of the function 𝜗(𝑥) is given by (2.3). Since tan(𝜃(𝑥))=𝜗(𝑥), where 𝜗(𝑥) is the derivative of 𝜗(𝑥), we have using (2.4) that 𝛿tan𝜋/2=𝜗2𝜗1𝛿(𝑥),(2.9) where 𝛿 is the Dirac generalized function.

3. Some Useful Lemmas

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.

Lemma 3.1. Given 𝜏1>0, let it be denoted by 𝑆𝜏1 and 𝐻 the step and Heaviside generalized functions respectively. Then the following relations hold: 𝑆𝜏1(𝑥𝑐𝑡)𝐻(𝑥)𝑀𝑆𝜏1𝑆(𝑥𝑐𝑡),(3.1)𝜏1(𝑥+𝑐𝑡)𝐻(𝑥)0,(3.2) where 𝑀, 𝑐>0 are constants and 𝑡>0.

Proof. We have that 𝑆𝜏1(𝑥𝑐𝑡)=𝛿(𝑥𝑐𝑡+𝜏1)𝛿(𝑥𝑐𝑡𝜏1). From this there exists constant 𝑀 such that for 𝑡>0, 𝑐>0, and 𝑐𝑡𝜏1>0, we have 𝛿(𝑥𝑐𝑡+𝜏1)𝐻(𝑥)𝑀𝛿(𝑥𝑐𝑡+𝜏1) and 𝛿(𝑥𝑐𝑡𝜏1)𝐻(𝑥)𝑀𝛿(𝑥𝑐𝑡𝜏1), then (3.1) holds.
It possible to check that for 𝑡>0, 𝑐>0, and 𝑐𝑡+𝜏1<0, (3.2) holds.

Lemma 3.2. Given 𝜏1>0, let it be denoted by 𝑆𝜏1 the step generalized functions. Then the following relations hold: 𝑆𝜏1𝑆(𝑥𝑐𝑡)𝛿(𝑥)0,(3.3)𝜏1(𝑥+𝑐𝑡)𝛿(𝑥)0,(3.4) for 𝑡>0 and 𝑐>0.

Proof. We have that 𝑆𝜏1(𝑥𝑐𝑡)=𝐻(𝑥𝑐𝑡+𝜏1)𝐻(𝑥𝑐𝑡𝜏1), 𝛿(𝑥)𝐻(𝑥𝑐𝑡+𝜏1)0, and 𝛿(𝑥)𝐻(𝑥𝑐𝑡+𝜏1)0 for 𝑡>0 and 𝑐>0; thus, (3.3) holds. Analogously, it is possible to verify that (3.4) holds.

The following propositions are useful.

Lemma 3.3. Given 𝜏1>0, 𝑐>0, and 𝑡 such that 𝑡>𝜏1/𝑐, let it be denoted by 𝑆𝜏1 and 𝑆𝜏1 the step soliton and its derivative generalized functions, respectively. Then the following relations hold:(i)𝑆2𝜏1𝑆𝜏1,(ii)𝑆𝜏1(𝑥𝑐𝑡)𝑆𝜏1(𝑥+𝑐𝑡)0,(iii)𝑆𝜏1(𝑥𝑐𝑡)𝑆𝜏1(𝑥+𝑐𝑡)0.

Proof. We prove here that (ii) the others are similar. We have that 𝑆𝜏1(𝑥𝑐𝑡)=𝐻(𝑥𝑐𝑡+𝜏1)𝐻(𝑥𝑐𝑡𝜏1) and 𝑆𝜏1(𝑥+𝑐𝑡)=𝛿(𝑥+𝑐𝑡+𝜏1)𝛿(𝑥+𝑐𝑡𝜏1), where 𝛿 and 𝐻 are the Dirac and Heaviside generalized function. It is possible to check that for 𝑡>𝜏1/𝑐, the delta soliton of the 𝑆𝜏1(𝑥𝑐𝑡) stays in the null part of step soliton 𝑆𝜏1(𝑥𝑐𝑡), so (ii) holds.

Lemma 3.4. Given 𝜏1>0, 𝑐>0, and 𝑡 such that 𝑡>𝜏1/𝑐, let it be denoted by 𝑆𝜏1 and 𝛿1 the step soliton and microscopic generalized functions, respectively. Then the following relations hold:(i)𝑆𝜏1(𝑥+𝑐𝑡)𝛿1(𝑥)0,(ii)𝑆𝜏1(𝑥𝑐𝑡)𝛿1(𝑥)0.

4. The Flat Bottom Case

4.1. Nonlinear Effect

We consider the so-called shallow water equations in one dimension as given in [6]. Here, we put these equations in the sense of associations of Colombeau as follows: 𝑡+(𝑢)𝑥0,(4.1)(𝑢)𝑡+12𝑢2𝑥+𝑔𝑥0,(4.2) where is the height of water, 𝑢 is the velocity, and 𝑔 is the gravity constant. This model is relevant even to deep water as long as the velocity stays constant on the thickness of the water layer, otherwise this model corresponds to a damped model since the velocity is averaged which can be deduced, as seen easily; by using the Cauchy Schwartz inequality. We split the height of water as =𝑜+𝜂, where 𝑜 is the bottom depth, and 𝜂 is the surface elevation relative to the fixed depth 𝑜 (which is the case in Figure 2 if the angle in respect to the OX axis is zero, i.e., 𝜃=0). As in [24], we take the following.

910659.fig.002
Figure 2: Schematic diagram of a solitary wave propagating over a mild slope bottom.

Assumption 4.1. 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 𝜏1, let us denote by 𝑆𝜏1 the derivative of the step soliton generalized function 𝑆𝜏1. We interpret (4.1) and (4.2) in the sense of association, that is, we seek the analog of classical weak solutions (see [26, 27, 47, 50]). We are going to seek solutions of the system (4.1) and (4.2) in the form of 𝜆𝑆𝜏1 where 𝑆𝜏1 is a step soliton generalized function. The following theorem holds.

Theorem 4.2. It is assumed that solitons of the system (4.1) and (4.2) are given by𝜂=𝜆𝑆𝜏1(𝑥𝑋(𝑡)),(4.3a)𝑢=𝑢𝑜𝑆𝜏1(𝑥𝑋(𝑡)),(4.3b)for a given 𝜏1, where 𝜆 and 𝑢𝑜 are constants representing the amplitude of surface elevation and particle velocity, respectively, and =𝑜+𝜂, where 𝑜 is a fixed real number. Here, 𝑋(𝑡) is the trajectory where the singularity travels and 𝑐=𝑋(𝑡) denotes the soliton velocity. Assuming that 𝜆 is known, then the wave velocity 𝑐 and amplitude of particle velocity 𝛼 are given by𝑢𝑜=𝜆𝑔𝑜,+𝜆/2(4.4a)𝑐=𝑜+𝜆𝑔𝑜.+𝜆/2(4.4b)

Proof. Using that =𝑜+𝜂 and substituting (4.3a) and (4.3b) in (4.1) with 𝜉=𝑥𝑋(𝑡), we obtain 𝜆𝑋𝑆𝜏1(𝜉)+𝑢𝑜𝜆𝑆𝜏1(𝜉)𝑆𝜏1(𝜉)+𝑢𝑜𝜆𝑆𝜏1(𝜉)𝑆𝜏1(𝜉)+𝑜𝑢𝑜𝑆𝜏1(𝜉)0.(4.5) Now, using that 𝑆𝜏1(𝜉)𝑆𝜏1(𝜉)=(1/2)(𝑆2𝜏1(𝜉)), we have 𝜆𝑋𝑆𝜏1(𝜉)+𝑢𝑜𝜆𝑆2𝜏1(𝜉)+𝑜𝑢𝑜𝑆𝜏1(𝜉)0.(4.6) Finally, from the fact that 𝑆2𝜏1(𝜉)𝑆𝜏1(𝜉), we deduce that 𝜆𝑋𝑆𝜏1(𝜉)+𝑢𝑜𝜆𝑆𝜏1(𝜉)+𝑜𝑢𝑜𝑆𝜏1(𝜉)0.(4.7) Since 𝑆𝜏1(𝜉) is not associate to null generalized function, such above equation implies that𝑋𝜆+𝑢𝑜𝜆+𝑜𝑢𝑜𝑋=0,(4.8a)=𝑢𝑜𝜆+𝑜𝜆.(4.8b)Since that right hide side of (4.8b) is a constant, then the trajectory of the singularity is the straight line rect, that is, 𝑋𝑢(𝑡)=𝑜𝜆+𝑜𝜆𝑡+𝐾,(4.9) where 𝐾 is a constant. As a consequence, the soliton velocity is given by 𝑐=𝑋𝑢(𝑡)=𝑜𝜆+𝑜𝜆.(4.10) Now, substituting (4.3a) and (4.3b) in (4.2) and using again the fact that 𝑆𝜏1𝑆𝜏1=(1/2)(𝑆2𝜏1), we obtain 𝑢𝑜𝑋𝑆𝜏1+12𝑢2𝑜𝑆2𝜏1+𝑔𝜆𝑆𝜏10,(4.11) or equivalently, 𝑋𝑢𝑜+𝑢2𝑜2𝑆+𝑔𝜆𝜏10.(4.12) Since 𝑆𝜏1 is not associate to null generalized function, from (4.12), we obtain 𝑋𝑢𝑜+𝑢2𝑜2+𝑔𝜆=0.(4.13) Substituting (4.10) in (4.13), we have 2𝑢2𝑜𝑜+𝜆+2𝑔𝜆2+𝜆𝑢2𝑜=0.(4.14) From (4.14) we obtain (4.4a), and from (4.4a) and (4.10) we obtain that (4.4b) holds.

Remark 4.3. The choice of the particle velocity 𝑢 as a product by the step generalized function (see (4.3b)) like the free surface stays in concordance which linear wave theory, see as an example [45, 51].

Remark 4.4. Taking off the amplitude wave 𝜆 from (4.4b) and substituting in (4.4a) we obtain 𝑢𝑜=𝑢𝑜𝑜𝑐𝑢𝑜𝑔𝑜+𝑢𝑜𝑜/𝑐𝑢𝑜.(4.15) Thus, we obtain a close system of equations with (4.4a) and (4.4b), and (4.15), which allows to estimate the wave celerity, velocity particle, and wave amplitude (𝑐,𝑢𝑜,𝜆) by using quasi-Newton method, for example.

Theorem 4.2 has an immediate practical sense: the trajectory of the singularity is linear for the case of planar bottom with the system of (4.1) and (4.2).

Let us denote 𝜎=𝜆/𝑜, 𝜇=(𝑘)2, where 𝑘 is number wave, as the nonlinear and dispersive parameters, respectively. From now, we compared the formulas obtained with previous solutions. To do so, we compared the wave celerity of different formulations (see [52]). It is possible to rewrite the wave celerity (4.4b) as follows: 𝑐=𝑔𝑜1+𝜆/𝑜1+𝜆/2𝑜11+𝜆/2𝑜.(4.16) Equation (4.16) for small nonlinear parameter 𝜎1 holds, 𝑐=𝑔𝑜31+4𝜆𝑜5𝜆32𝑜2+7𝜆128𝑜3𝜆+𝑂𝑜4.(4.17) Formula (4.17) is similar to those obtained in [6, 5355] which depends on the nonlinear parameter 𝜎. It is possible to check that the difference of the formula (4.4b) in respect of those obtained in the above-cited review has order 𝜎. In particular, we consider the wave celerity obtained in ([6, page 463]), that is, 𝑐1=(3𝑔(𝑜+𝜆)2𝑔𝑜)=𝑔𝑜(3(1+𝜆/𝑜)1/22), which for small 𝜎 holds as follow, 𝑐1=𝑔𝑜31+2𝜆𝑜38𝜆𝑜2+3𝜆16𝑜3𝜆+𝑂𝑜4.(4.18) It is possible to verify that the quotient between (4.17) and (4.18) is approximately |𝑐|/|𝑐1|1(3/4)𝜎+(43/32)𝜎2+(311/128)𝜎3. Thus, we obtain good matches (maximum difference of less than 10 percent) for 𝜎<0.4 (see Figure 3(a)).

fig3
Figure 3: 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 𝜎=𝑂(𝜇).

Also, when 𝜎=𝑂(𝜇), the formula for the wave celerity (4.17) is similar to those obtained in [8, 25, 56, 57]. In particular, the quotient in respect to the classical dispersion linear (Airy’s wave celerity): 𝑐2=𝑔𝑜tanh(𝑘)=𝑘𝑔𝑜116(𝑘)2+19360(𝑘)4553024(𝑘)6+𝑂(𝑘)8(4.19) is approximately |𝑐|/|𝑐1|1+(11/12)𝜇(9/160)𝜇2(53/17280)𝜇3. Thus, we obtain maximum difference of less than 10 percent for 𝜇<0.1 (see Figure 3(b)). This small range of good matches is expected because in the deduction of (4.4a) and (4.4b), we do not consider the dispersive effect in shallow water equations.

4.2. Nonlinear and Dispersive Effects

We consider the following so-called shallow water equations with dispersive effect in one dimension as given in [8]: 𝜂𝑡+𝑢𝑥+(𝜂𝑢)𝑥+1𝛼+33𝑢𝑥𝑥𝑥=0,(𝑢)𝑡+𝑔𝜂𝑥+12𝑢2𝑥+𝛼2𝑢𝑡𝑥𝑥=0,(4.20) where is the height of water, 𝑢 is the velocity, 𝑔 is the gravity constant, and 𝛼=(1/2)(𝑧𝛼/)2+(𝑧𝛼/) at reference depth 𝑧𝛼. We assume here that the bottom is constant, that is, =𝑜. But with the method presented in this paper, it is possible to obtain generalized solutions regarding variable bottom.

The following theorem holds.

Theorem 4.5. It is assumed that solitons of the system (4.20) are given by 𝜂=𝜆𝑆𝜏1(𝑘𝑥𝜔𝑡),𝑢=𝑢𝑜𝑆𝜏1(𝑘𝑥𝜔𝑡),(4.21) for a given 𝜏1, where 𝜆 and 𝑢𝑜 are constants representing the amplitude of surface elevation and particle velocity, respectively, and =𝑜+𝜂, where 𝑜 is a fixed real number. Here, 𝑘,𝜔 are the wave number and frequency, respectively. Then the following equalities hold: 𝑢𝑜=𝜆𝑔𝑜𝑜1𝜈(1+𝜎/2)+2/2𝛼𝜇(1+𝜎)+𝜈1,𝑢(𝛼+(1/3)𝜇𝑘)(4.22)𝜔=𝑜𝑘𝜆𝜆+𝑜+𝜈11𝛼+3𝜇𝑜,(4.23) where 𝜈1, 𝜈2 are arbitrary constants and 𝜎 and 𝜇 are the nonlinear and dispersive parameters, respectively.

Proof. Since the proof is similar to Theorem 4.2, we present a summary here. The idea of the proof consists in substituting the generalized function (4.21) in the system (4.20). By using the relations 𝑆2𝜏1(𝜉)𝑆𝜏1(𝜉) and 𝑆𝜏1(𝜉)𝑆𝜏1(𝜉)=(1/2)(𝑆2𝜏1(𝜉)), 𝜉=𝑘𝑥𝜔𝑡 and after several operations, we obtain 𝜔𝜆+𝑢𝑜𝑘𝑜𝑆+𝜆𝜏11(𝜉)+𝛼+3𝑘33𝑜𝑢𝑜𝑆𝜏1(𝜉)0,𝑢𝑜1𝜔+2𝑢2𝑜𝑆𝑘+𝑔𝜆𝑘𝜏1(𝜉)𝛼𝑘22𝑜𝑢𝑜𝜔𝑆𝜏1(𝜉)0.(4.24) Finally, taking a representant 𝑅(𝜖,𝜉)=𝑎1+𝑎2𝜖𝜉+𝑎3𝜖2𝜉2+𝑎4𝜖3𝜉3+𝑂(𝜖4𝜉4) of 𝑆𝜏1 and using the Definition 2.2, we obtain that there exist constants 𝜈1,𝜈2 such that 𝑢𝜔=𝑜𝑘𝑜+𝜆𝜆+𝜈1(𝛼+(1/3))𝑘3𝑢𝑜3𝑜𝜆,(4.25)𝑢𝑜1𝜔+2𝑢2𝑜𝑘+𝑔𝜆𝑘𝜈2𝛼𝑘22𝑜𝜔𝑢𝑜=0.(4.26) Combining (4.25) and (4.26), we obtain (4.22) and (4.23).

Remark 4.6. Taking 𝜈1=𝜈2=0 in (4.25) and (4.26), that is, neglecting the dispersive effects, it is possible to verify that (4.22) and (4.23) are the same as that (4.4a) and (4.4b) in Theorem 4.2 (the nonlinear effect alone), which indicates that the calculations are consistent.

From (4.23), we can deduce the wave celerity as 𝑐=𝑔𝑜1+𝜎+𝜈1(𝛼+(1/3))𝜇(1+𝜎/2)+(1/2)𝜈2𝛼𝜇(1+𝜎)+𝜈1(𝛼+(1/3))𝜇𝜇.(4.27) The expression (4.27) is similar to those obtained in [29]. In the following, we verify the similitude of formula (4.27) with Airy’s wave celerity. In the simulation we assume that 𝜎=𝑂(𝜇) and 𝑜=1. Also we take the value of parameter 𝛼=0.39 from [8]. In Figures 4(a) and 4(b), we present the quotient of the wave celerity (4.27) with Airy’s wave celerity, depending on the dispersive parameter 𝜇 from shallow water (0<𝜇<𝜋/10) to transitional (𝜋/10<𝜇<𝜋). An optimum value of the parameter (𝜈1,𝜈2)=(4.17,1.7) for the range, 0<𝜇<2.5 with 𝜎=𝜇, by minimizing the sum of the relative difference between the two wave celerity studies was obtained here. We can see that several pairs of optimum parameters (𝜈1,𝜈2) produce good matches with greater interval which is better than the nonlinear case (see Figures 4(a) and 4(b)).

fig4
Figure 4: Comparison of wave celerity for dispersive and nonlinear of the same order 𝜎=𝑂(𝜇).

5. A Discontinuity Bottom Case

In this section, we studied the case in which a soliton crosses a bottom discontinuity (see Figure 5). Seeking the solution of shallow water equation requires some useful lemmas that were proved in Section 3. These propositions contain the key results of the product of generalized functions that appear in the algebraic operations when generalized solution is searched.

910659.fig.005
Figure 5: Schematic diagram of a solitary wave propagating over a discontinuity bottom.
5.1. Generalized Solution

Following the same idea as in the previous section, we obtain a generalized solution of shallow water equation stated in [6] as in this case one takes into account friction and slope of the bottom 𝑡+(𝑢)𝑥=0,(5.1)(𝑢)𝑡+12𝑢2𝑥+𝑔𝑥=𝑔𝑆𝐶𝑓𝑢2,(5.2) where 𝑔=𝑔cos(𝜃), 𝑆=tan(𝜃) with bottom slope 𝜃 (see Figure 2). Here, 𝐶𝑓 denotes the friction coefficient. Neglecting friction, (5.2) in generalized sense of association is given by 𝑡+(𝑢)𝑥0,(5.3)(𝑢)𝑡+12𝑢2𝑥+𝑔𝑥𝑔𝑆.(5.4) Now, we assume that the depth has a jump in the bottom (see Figure 5). In this case, the bottom can be written as 𝑜(𝑥)=1+21𝐻(𝑥),(5.5) where 𝐻 is the Heaviside generalized function, and Δ=21 and 1, 2 are constants.

5.1.1. A Case of Single Soliton

Given 𝜏1>0, we find a generalized solution of system (5.3) and (5.4) as 𝜂(𝑥,𝑡)=𝜆𝑆𝜏1(𝑥𝑋(𝑡)),𝑢(𝑥,𝑡)=𝛼𝑆𝜏1(𝑥𝑋(𝑡)),(5.6) where 𝑋(𝑡) is that trajectory of the singularities, and 𝑆𝜏1 is the step generalized function. We assume that at time 𝑡=0, the generalized solution is known, that is, 𝜂(𝑥,0)=𝜆𝑆𝜏1(𝑥),𝑢(𝑥,0)=𝑢𝑜𝑆𝜏1(𝑥),(5.7) where 𝜆 and 𝑢𝑜 are known constants. The following theorem holds.

Theorem 5.1. It is assumed that solitons of the system (5.3) and (5.4) are given by 𝜂=𝜆𝑆𝜏1(𝑥𝑋(𝑡)),𝑢=𝑢𝑜𝑆𝜏1(𝑥𝑋(𝑡)),(5.8) for a given 𝜏1, where 𝜆 and 𝑢𝑜 are constants representing the amplitude of surface elevation and particle velocity, respectively, and =𝑜+𝜂, where 𝑜 is given in (5.5). Here, 𝑋(𝑡) is the trajectory where the singularity travels and let it be denoted by 𝑐1=𝑋(𝑡) for 𝑥<0 and 𝑐2=𝑋(𝑡) for 𝑥>0 the soliton velocity. Assuming that 𝜆 is known, then the soliton velocities 𝑐1, 𝑐2 are given by 𝑐1=𝑢𝑜1+𝜆𝜆,𝑐2=𝑢𝑜2+𝜆𝜆.(5.9)

Proof. Substituting (5.6) and (5.5) in (5.3) with 𝜉=𝑥𝑋(𝑡), we obtain 𝑋𝜆𝑆𝜏1𝑢(𝜉)+𝑜𝑆𝜏1(𝜉)Δ𝛿(𝑥)+𝜆𝑆𝜏1+(𝜉)1+Δ𝐻(𝑥)+𝜆𝑆𝜏1𝑢(𝜉)𝑜𝑆𝜏1(𝜉)0.(5.10) Using that 𝑆𝜏1(𝜉)𝑆𝜏1(𝜉)(1/2)(𝑆2𝜏1(𝜉)), Lemma 3.2 and Lemma 3.3(i), that we have from (5.10) 𝑋1𝜆+2𝑢𝑜𝜆+𝑢𝑜1+12𝑢𝑜𝜆+𝑢𝑜𝑆Δ𝐻(𝑥)𝜏1(𝜉)0.(5.11) Since 𝑆𝜏1(𝜉) is not associate to null generalized function, we obtain 𝑋1𝜆+2𝑢𝑜𝜆+𝑢𝑜1+12𝑢𝑜𝜆+𝑢𝑜Δ𝐻(𝑥)=0.(5.12) From (5.12), we obtain (5.9).

Remark 5.2. Theorem 5.1 indicates that the trajectory of the singularity of one soliton that passes by the discontinuity point in the bottom consist in a cone. Moreover, the velocity of the soliton is constant and different in both sides of the jump. This suggests from the physical point of view that happened, a rectification of the soliton and velocity only depends on the depth.

5.1.2. A Case of Two Solitons

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 5.2, we assume that the velocity of the solitons is constant.

Given 𝜏1>0, we find a generalized solution of system (5.3) and (5.4) as 𝜂(𝑥,𝑡)=𝜆1𝑆𝜏1𝑥𝑐1𝑡+𝜆2𝑆𝜏1𝑥+𝑐2𝑡,𝑢(𝑥,𝑡)=𝑢𝑜1𝑆𝜏1𝑥𝑐1𝑡+𝑢𝑜2𝑆𝜏1𝑥+𝑐2𝑡,(5.13) where 𝑐1, 𝑐2 are constants, and 𝑆𝜏1 is the step generalized function. We assume that at time 𝑡=0, the generalized solution is known, that is, 𝜂𝜆(𝑥,0)=1+𝜆2𝑆𝜏1(𝑥)=𝜆𝑆𝜏1𝑢(𝑥),𝑢(𝑥,0)=𝑜1+𝑢𝑜2𝑆𝜏1(𝑥)=𝑢𝑜𝑆𝜏1(𝑥),(5.14) where 𝜆=𝜆1+𝜆2 and 𝑢𝑜=𝑢𝑜1+𝑢𝑜2 are considered as constants. In this case, we consider the discontinuous bottom as in (5.5). The following theorem holds.

Theorem 5.3. For given 𝜏1>0, let it be assumed that a generalized solution of (5.3) and (5.4) is given by (5.13) with bottom depth given in (5.5). Assuming that the amplitudes 𝜆 and 𝑢𝑜 are known, then the wave velocities 𝑐1 and 𝑐2, the amplitude of particle velocity 𝑢𝑜2, and the amplitude 𝜆2 of reflected wave satisfy on 𝑡>min{𝜏1/𝑐1,𝜏1/𝑐2} the following algebraic equations: 𝑐1𝜆𝜆2+𝑢𝑜𝑢𝑜2𝜆𝜆2+1𝜆+𝑀Δ=0,(5.15)2𝑐2+𝑢𝑜2𝜆2+1=0,(5.16)𝑐1𝜆𝜆2+12𝑢𝑜𝑢𝑜22+𝑔𝜆𝜆2𝜆=0,(5.17)2𝑐2+12𝑢2𝑜2+𝑔𝜆2=0,(5.18) where 𝑀 is a constant.

Proof. Denote that by 𝜉1=𝑥𝑐1𝑡 and 𝜉2=𝑥+𝑐2𝑡, we have 𝜂(𝑥,𝑡)=𝜆𝜆2𝑆𝜏1𝜉1+𝜆2𝑆𝜏1𝜉2,𝑢𝑢(𝑥,𝑡)=𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝑢𝑜2𝑆𝜏1𝜉2,(𝑥,𝑡)=𝑜(𝑥)+𝜂(𝑥,𝑡).(5.19) Now, substituting (5.19) in (5.3) we obtain 𝑜(𝑥)𝑡+𝑐1𝜆𝜆2𝑆𝜏1𝜉1+𝑐2𝜆2𝑆𝜏1𝜉2+𝑢𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝑢𝑜2𝑆𝜏1𝜉21+Δ𝐻(𝑥)𝑥+𝑢𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝑢𝑜2𝑆𝜏1𝜉2𝜆𝜆2𝑆𝜏1𝜉1+𝜆2𝑆𝜏1𝜉2𝑢+(𝑥,𝑡)𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝑢𝑜2𝑆𝜏1𝜉20.(5.20) Now, using that 𝑆𝜏1𝑆𝜏1=(1/2)(𝑆2𝜏1) and from Lemma 3.3 that 𝑆𝜏1(𝜉1)𝑆𝜏1(𝜉2)0, 𝑆𝜏1(𝜉2)𝑆𝜏1(𝜉1)0, we have 𝑐1𝜆𝜆2𝑆𝜏1𝜉1+𝑐2𝜆2𝑆𝜏1𝜉2𝑢+Δ𝑜𝑢𝑜2𝑆𝜏1𝜉1𝑆(𝑥)+𝑢𝑜2𝑆𝜏1𝜉2𝑆+1(𝑥)2𝑢𝑜𝑢𝑜2𝜆𝜆2𝑆2𝜏1𝜉1+12𝜆2𝑢𝑜2𝑆2𝜏1𝜉2+1𝑢𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝑢𝑜2𝑆𝜏1𝜉2𝑢+Δ𝑜𝑢𝑜2𝑆𝜏1𝜉1𝐻(𝑥)+𝑢𝑜2𝑆𝜏1𝜉2+1𝐻(𝑥)2𝜆𝜆2𝑢𝑜𝑢𝑜2𝑆2𝜏1𝜉1+𝜆2𝑢𝑜2𝑆2𝜏1𝜉20.(5.21) From Lemma 3.1, we have 𝑆𝜏1(𝜉1)𝐻(𝑥)𝑀𝑆𝜏1(𝜉1) and 𝑆𝜏1(𝜉2)𝐻(𝑥)0 for some constant 𝑀 and for 𝑡,𝑐>0. Also, from Lemma 3.2 we have 𝑆𝜏1(𝜉1)𝛿(𝑥)0 and 𝑆𝜏1(𝜉2)𝛿(𝑥)0 for 𝑐1,𝑐2,𝑡>0, and since 𝑆2𝜏1𝑆𝜏1 (see Lemma 3.3(i)), we obtain 𝑐1𝜆𝜆2+12𝑢𝑜𝑢𝑜2𝜆𝜆2+1𝑢𝑜𝑢𝑜2𝑢+𝑀Δ𝑜𝑢𝑜2+12𝜆𝜆2𝑢𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝜆2𝑐2+12𝑢𝑜2𝜆2+1𝑢𝑜2+12𝜆2𝑢𝑜2𝑆𝜏1𝜉20.(5.22) Analogously, substituting (5.19) in (5.4), we obtain 𝑐1𝜆𝜆2𝑆𝜏1𝜉1+𝑐2𝑢𝑜2𝑆𝜏1𝜉2+𝑢𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝑢𝑜2𝑆𝜏1𝜉2𝑢𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝑢𝑜2𝑆1𝜉2+𝑔1+Δ𝐻(𝑥)𝑥+𝑔𝜆𝜆2𝑆𝜏1𝜉1+𝜆2𝑆𝜏1𝜉2𝑔tan(𝜃).(5.23) Now, from Lemma 3.3 (i), we have 𝑆2𝜏1𝑆𝜏1. Also from Lemma 3.3(ii)(iii), we have 𝑆𝜏1(𝜉1)𝑆𝜏1(𝜉2)0 and 𝑆𝜏1(𝜉1)𝑆𝜏1(𝜉2)0 for 𝑡>min{𝜏1/𝑐1,𝜏1/𝑐2}. Since 𝑆𝜏1𝑆𝜏1=(1/2)(𝑆2𝜏1), 𝑆𝜏1(𝜉1)𝛿10, 𝑆𝜏1(𝜉2)𝛿10 (Lemma 3.4(i)(ii)), we obtain 𝑔Δ𝐻+𝑐1𝜆𝜆2+12𝑢𝑜𝑢𝑜22+𝑔cos(𝜃)𝜆𝜆2𝑆𝜏1𝜉1+𝑢𝑜2𝑐1+12𝑢2𝑜2+𝑔cos(𝜃)𝜆2𝑆𝜏1𝜉2𝑔tan(𝜃).(5.24) Finally, using that cos(𝜃)1𝛿1(𝑥) and tan(𝜃)Δ𝛿, where 𝜃 is the angle in respect to axis 𝑂𝑋 (see (2.6) and (2.9)) and using Lemma 3.4, we have 𝑔Δ𝛿+𝑐1𝜆𝜆2+12𝑢𝑜𝑢𝑜22+𝑔𝜆𝜆2𝑆𝜏1𝜉1+𝑢𝑜2𝑐1+12𝑢2𝑜2+𝑔𝜆2𝑆𝜏1𝜉2𝑔Δ𝛿,(5.25) or equivalently, 𝑐1𝜆𝜆2+12𝑢𝑜𝑢𝑜22𝑢+𝑔𝑜𝑢𝑜2𝑆𝜏1𝜉1+𝑢𝑜2𝑐1+12𝑢2𝑜2+𝑔𝜆2𝑆𝜏1𝜉20.(5.26) Because that the generalized function of the left hand of (5.22) and (5.26) is equivalent to zero, it is necessary that the coefficient of 𝑆1(𝜉1) and 𝑆1(𝜉2) must be zero. So, the system of (5.15)–(5.18) holds.

Remark 5.4. Although Theorem 5.3 was obtained for a discontinuity in the bottom, it is not difficult to generalize this result for any type of bottom. To do so, any geometric of the bottom can be approximated by step functions, and then theorem can be used locally.

Remark 5.5. In Theorem 5.3, we assume that 𝑡>min{𝜏1/𝑐1,𝜏1/𝑐2}. If we relax this hypothesis, that is, to obtain the generalized solution on 0<𝑡<min{𝜏1/𝑐1,𝜏1/𝑐2}, we have that, following relations hold: 𝑆𝜏1𝜉1𝑆𝜏1𝜉2𝛿𝑥𝑐𝑡𝜏1,𝑆𝜏1𝜉1𝑆𝜏1𝜉2𝛿𝑥+𝑐𝑡+𝜏1,(5.27) where 𝛿 is the Dirac generalized function. The product of generalized functions (5.27) was taken as null in the proof of Theorem 5.3. In the contrary case, it is possible to check that in the proof (similar to Theorem 5.3), a new equation arises due to the coefficients of 𝑆𝜏1(𝜉1)𝑆𝜏1(𝜉2) and 𝑆𝜏1(𝜉1)𝑆𝜏1(𝜉2), which is 𝑢𝑜𝑢𝑜2𝑢𝑜2=0.(5.28) Equation (5.28) has two solutions which are 𝑢𝑜=𝑢𝑜2 or 𝑢𝑜2=0. More physical sense has the solution 𝑢𝑜2=0, which means that for 𝑡<min{𝜏1/𝑐1,𝜏1/𝑐2}, the reflected effect of wave velocity particles is not starting yet. In this point, a rise of the wave amplitude near the leading edge of the discontinuous point occurs due to the shallow effect [51]. In that case it is possible to verify that system (5.15)–(5.18) reduces to the system of equations 𝑐1𝜆+𝑢𝑜𝜆+1+𝑀Δ=0,𝑐11𝜆+𝑔𝜆+2𝑢2𝑜=0.(5.29) Equation (5.29) for known 𝜆 has the explicit solutions: 𝑢𝑜1,2=1±+𝜆+𝑀Δ1+𝜆+𝑀Δ22𝑔𝜆,𝑐11,2=𝑢𝑜1,21+𝜆+𝑀Δ𝜆,(5.30) with 𝑐2=𝜆2=𝑢𝑜2=0. However, taking off the amplitude wave 𝜆 of (5.29) and equaling it, we obtain 𝑢𝑜/2𝑔𝑐1=1+𝑀Δ𝑐1𝑢𝑜.(5.31) Now, solving a close system (5.29), and (5.31), we obtain (𝑐1,𝑢𝑜,𝜆), that is, wave celerity, particle velocity, and wave amplitude, respectively.

6. Numerical Calculation of the Generalized Solution

In this section, we show a numerical procedure to find the unknown parameters 𝜆2 and 𝑢𝑜2 which are solution of the system of (5.15)–(5.18). In practical terms to determine those parameters means to calculate the amplitude of the step Soliton when it passes through a point of discontinuity in the bottom. The method consists in reducing the set of four equations to two by eliminating the unknowns 𝑐1 and 𝑐2. The following lemma holds.

Lemma 6.1. Let it be assumed that a generalized solutions of (5.3) and (5.4) is given by (5.13) with bottom depth given in (5.5). Assuming that 𝜆 and 𝑢𝑜 are known, then the amplitude of particle velocity 𝑢𝑜2 and the amplitude 𝜆2 of the reflected wave satisfy 12𝑢2𝑜2𝜆2𝑢𝑜2+𝑔𝜆2𝑢𝑜211=0,(6.1)2𝑢𝑜𝑢𝑜22𝜆𝜆2𝑢𝑜𝑢𝑜2+𝑔𝜆𝜆2𝑢𝑜𝑢𝑜21+𝑀Δ=0,(6.2) where 𝑀 is a constant.

Proof. Equation (6.1) follows from (5.18) minus (5.16). Equation (6.2) follows from (5.17) minus (5.15).

Let us denote 𝐺1𝑢𝑜2,𝜆2=12𝑢2𝑜2𝜆2𝑢𝑜2+𝑔𝜆2𝑢𝑜21,𝐺2𝑢𝑜2,𝜆2=12𝑢𝑜𝑢𝑜22𝜆𝜆2𝑢𝑜𝑢𝑜2+𝑔𝜆𝜆2𝑢𝑜𝑢𝑜21.+𝑀Δ(6.3) Now, to find the zeros of (6.1) and (6.2) is equivalent to find the zeros of the application 𝑢𝐺𝑜2,𝜆2𝐺1𝑢𝑜2,𝜆2,𝐺2𝑢𝑜2,𝜆2,(6.4) in the region 𝐵={(𝑢𝑜2,𝜆2)0<𝑢𝑜2<𝑢𝑜 and 0<𝜆2<𝜆}. To do so, it is possible to use the quasi-Newton method.

Remark 6.2. Taking 𝑢𝑜2=0, 𝜆2=0, and Δ=0, that is, the flat bottom case, then it is possible to verify that (6.1) and (6.2) is the same as the flat bottom case (4.4a) and (4.4b), which indicates that the calculation in the discontinuous bottom case is consistent.

7. Numerical Examples

In this section, we show that the generalized solutions with physical sense can be obtained. To do so, the constant 𝑀 in the system of (5.15)–(5.18) can be adjusted such that generalized solutions represent appropriately the theoretical and experimental data.

The initial values of quasi-Newton method for solving (6.1)-(6.2) are taken by using the formula for planar bottom case; that is, assuming the wave celerity is known from (4.4a) and (4.4b), we obtain 𝑔𝜆2+(2𝑔𝑜(𝑐2/2))𝜆+(𝑔2𝑜𝑜𝑐2)=0. It is possible to check that the positive root of the above equation produces a wave amplitude with reasonable value.

In [45, 58] was used the theoretical amplitude of soliton 𝜆1 in the impermeable case with a discontinuity bottom which was deduced in [45], which is 𝜆11/4=𝜆1/4+0.08356𝜐𝑔1/222𝑥2,(7.1) where 𝜆 is the initial amplitude, 𝑥 is the distance traveled by the soliton wave, and 𝜐 is the kinematic viscosity of the fluid. In [40], numerical results solving the Navier-Stokes equation match with the above theoretical result. The formula (7.1) to prove that the soliton generalized solution approximates the theoretical result is used in this paper.

We take the example described in [40] which considered the discontinuity bottom as 1=80 cm, 2=40 cm, and the initial amplitude 𝜆=4 cm. The theoretical result for this case using the formula (7.1) is compared with numerical solution of the system of (5.15)–(5.18). To approximate the theoretical solution, we present the generalized solution assuming that the constant 𝑀 in system of (6.1)-(6.2) depends on 𝑥/2, that is, 𝑀=𝑀(𝑥/2). This assumption enables us to show that the solution of (5.15)–(5.18) can reproduce well several amplitude step soliton values above the break point. We seek the values of the 𝑀(𝑥/2) that better adjusted the theoretical amplitude in (7.1) (see Figure 6). To do so, we use the solver fmincon.m in MATLAB 7.0. In Figure 6 is shows the theoretical and predicted step soliton amplitude when pass on a discontinuous depth point (𝑥=0).

910659.fig.006
Figure 6: Theoretical and predicted amplitude wave of the soliton propagating over a discontinued bed. The point 𝑥=0 coincides with the discontinuous bottom point.

In [46] was performed experiments to investigate the harmonic generation as periodic waves propagate over a submerged porous breakwater. Their experimental data will be used to test the validation of the present model equations for the wave and discontinuous bottom point interaction. We check that the generalized solution can reproduce well this experimental values.

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 𝑀 that appears in (5.15) can be estimated from several experimental runs looking for any regularity.

8. Conclusion

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.

Acknowledgments

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.

References

  1. R. F. Dressler, “New nonlinear shallow-flow equations with curvature,” Journal of Hydraulic Research, vol. 16, no. 3, pp. 205–222, 1978. View at Google Scholar · View at Scopus
  2. C. E. Synolakis, “The runup of solitary waves,” Journal of Fluid Mechanics, vol. 185, pp. 523–545, 1987. View at Google Scholar · View at Scopus
  3. J. G. Zhou, D. M. Causon, D. M. Ingram, and C. G. Mingham, “Numerical solutions of the shallow water equations with discontinuous bed topography,” International Journal for Numerical Methods in Fluids, vol. 38, no. 8, pp. 769–788, 2002. View at Publisher · View at Google Scholar · View at Scopus
  4. J. B. Keller, “Shallow-water theory for arbitrary slopes of the bottom,” Journal of Fluid Mechanics, vol. 489, pp. 345–348, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. I. J. Losada, M. D. Patterson, and M. A. Losada, “Harmonic generation past a submerged porous step,” Coastal Engineering, vol. 31, no. 1–4, pp. 281–304, 1997. View at Google Scholar · View at Scopus
  6. G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, NY, USA, 1974.
  7. A. A. Petrov, “Variational statement of the problem of liquid motion in a container of finite dimensions,” Journal of Applied Mathematics and Mechanics, vol. 28, no. 4, pp. 917–922, 1964. View at Google Scholar · View at Scopus
  8. O. Nwogu, “Alternative form of Boussinesq equations for nearshore wave propagation,” Journal of Waterway, Port, Coastal & Ocean Engineering, vol. 119, no. 6, pp. 618–638, 1993. View at Google Scholar · View at Scopus
  9. V. P. Maslov and G. A. Omelyanov, “Asymptotic soliton-like solutions of equations with small dispersion,” Russian Mathematical Surveys, vol. 36, no. 3, article 73, 1981. View at Google Scholar
  10. G. Elchahal, R. Younes, and P. Lafon, “The effects of reflection coefficient of the harbour sidewall on the performance of floating breakwaters,” Ocean Engineering, vol. 35, no. 11-12, pp. 1102–1112, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. F. Serre, “Contribution à létude des écoulements permanents et variables dans les canaux houille blanche,” Houille Blanche, no. 8, pp. 374–388, 1953. View at Google Scholar
  12. A. Gsponer, “A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics,” European Journal of Physics, vol. 30, no. 1, pp. 109–126, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. Miles and R. Salmon, “Weakly dispersive nonlinear gravity waves,” Journal of Fluid Mechanics, vol. 157, pp. 519–531, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. B. Savage and K. Hutter, “The motion of a finite mass of granular material down a rough incline,” Journal of Fluid Mechanics, vol. 199, pp. 177–215, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. D. Dutykh and D. Clamond, “Shallow water equations for large bathymetry variations,” Journal of Physics. A, no. 44, pp. 341–348, 2011. View at Publisher · View at Google Scholar
  16. F. Bouchut, A. Mangeney-Castelnau, B. Perthame, and J.-P. Vilotte, “A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 336, no. 6, pp. 531–536, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, vol. 84 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1984.
  18. G. H. Keulegan, “Gradual damping of solitary waves,” Journal of research of the National Bureau of Standards, no. 40, pp. 487–498, 1948. View at Google Scholar
  19. M. Calabrese, D. Vicinanza, and M. Buccino, “2D Wave setup behind submerged breakwaters,” Ocean Engineering, vol. 35, no. 10, pp. 1015–1028, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. A. Nachbin, “A terrain-following Boussinesq system,” SIAM Journal on Applied Mathematics, vol. 63, no. 3, pp. 905–922, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. A. E. Green and P. M. Naghdi, “Derivation of equations for wave propagation in water of variable depth,” Journal of Fluid Mechanics, vol. 78, no. 2, pp. 237–246, 1976. View at Google Scholar · View at Scopus
  22. D. Dutykh and F. Dias, “Dissipative Boussinesq equations,” Comptes Rendus. Mecanique, vol. 335, no. 9-10, pp. 559–583, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. R. Bernetti, V. A. Titarev, and E. F. Toro, “Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry,” Journal of Computational Physics, vol. 227, no. 6, pp. 3212–3243, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. H. A. Biagioni and M. Oberguggenberger, “Generalized solutions to the Korteweg-de Vries and the regularized long-wave equations,” SIAM Journal on Mathematical Analysis, vol. 23, no. 4, pp. 923–940, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. C. C. Mei, The Applied Dynamics of Ocean Surface Waves, Wiley, New York, NY, USA, 1983.
  26. J. F. Colombeau, Elementary Introduction to New Generalized Functions, vol. 113 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.
  27. J. F. Colombeau, “Generalized functions and infinitesimals,” Tech. Rep., 2006, http://arxiv.org/abs/math/0610264.
  28. S. Hamdi, W. H. Enright, Y. Ouellet, and W. E. Schiesser, “Method of lines solutions of the extended Boussinesq equations,” Journal of Computational and Applied Mathematics, vol. 183, no. 2, pp. 327–342, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. C.-J. Huang and C.-M. Dong, “On the interaction of a solitary wave and a submerged dike,” Coastal Engineering, vol. 43, no. 3-4, pp. 265–286, 2001. View at Publisher · View at Google Scholar · View at Scopus
  30. J. MeCowan, “On the solitary wave,” London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, vol. 5, no. 12, pp. 45–58, 1891. View at Google Scholar
  31. V. P. Maslov and G. A. Omelyanov, “Conditions of Hugoniot type for infinitely narrow solutions of the simple wave equation,” Sibirskii Matematicheskii Zhurnal, vol. 24, no. 5, pp. 787–795, 1983. View at Publisher · View at Google Scholar
  32. V. P. Maslov and V. A. Cupin, “Necessary conditions for the existence of infinitely narrow solitons in gas dynamics,” Soviet Physics, Doklady, vol. 246, no. 2, pp. 298–300, 1979. View at Google Scholar
  33. M. Nedeljkov, “Infinitely narrow soliton solutions to systems of conservation laws,” Novi Sad Journal of Mathematics, vol. 31, no. 2, pp. 59–68, 2001. View at Google Scholar · View at Zentralblatt MATH
  34. C. O. R. Sarrico, “New solutions for the one-dimensional nonconservative inviscid Burgers equation,” Journal of Mathematical Analysis and Applications, vol. 317, no. 2, pp. 496–509, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. C. O. R. Sarrico, “Entire functions of certain singular distributions and interaction of delta-waves in nonlinear conservation laws,” International Journal of Mathematical Analysis, vol. 4, no. 33-36, pp. 1765–1778, 2010. View at Google Scholar · View at Zentralblatt MATH
  36. F. Bouchut, A. Mangeney-Castelnau, B. Perthame, and J.-P. Vilotte, “A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 336, no. 6, pp. 531–536, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. R. Bernetti, V. A. Titarev, and E. F. Toro, “Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry,” Journal of Computational Physics, vol. 227, no. 6, pp. 3212–3243, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. F. Chazel, “Influence of bottom topography on long water waves,” M2AN. Mathematical Modelling and Numerical Analysis, vol. 41, no. 4, pp. 771–799, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. Z. Zhong and K. H. Wang, “Solitary wave interaction with a concentric porous cylinder system,” Ocean Engineering, vol. 33, no. 7, pp. 927–949, 2006. View at Publisher · View at Google Scholar · View at Scopus
  40. K. V. Karelsky, V. V. Papkov, A. S. Petrosyan, and D. V. Tsygankov, “Particular solutions of shallow-water equations over a non-flat surface,” Physics Letters. A, vol. 271, no. 5-6, pp. 341–348, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  41. C. Lin, T. C. Ho, S. C. Chang, S. C. Hsieh, and K. A. Chang, “Vortex shedding induced by a solitary wave propagating over a submerged vertical plate,” International Journal of Heat and Fluid Flow, vol. 26, no. 6, pp. 894–904, 2005. View at Publisher · View at Google Scholar · View at Scopus
  42. P. A. Madsen, H. B. Bingham, and H. A. Schäffer, “Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis,” The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences, vol. 459, no. 2033, pp. 1075–1104, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. J. B. Christopher and R. D. Dean, “Wave transformation by two-dimensional bathymetric anomalies with sloped transitions,” Coastal Engineering, vol. 50, no. 1-2, pp. 61–84, 2003. View at Publisher · View at Google Scholar · View at Scopus
  44. J. Fenton, “A ninth-oder solution for the solitary wave,” The Journal of Fluid Mechanics, vol. 58, no. 2, pp. 257–271, 1972. View at Google Scholar
  45. C. C. Mei, “Resonant reflection of surface water waves by periodic sandbars,” Journal of Fluid Mechanics, vol. 152, pp. 315–335, 1985. View at Google Scholar · View at Scopus
  46. M. A. Losada, A. C. Vidal, and R. Medina, “Experimental study of the evolution of a solitary wave at an abrupt junction,” Journal of Geophysical Research, no. 94, pp. 557–566, 1989. View at Google Scholar
  47. J. F. Colombeau, “Generalized functions and nonsmooth nonlinear problems in mathematics and physics,” Tech. Rep., 2006, http://arxiv.org/abs/math-ph/0612077.
  48. A. J. C. de Saint-Venant, “Thorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à lintroduction des mares dans leur lit,” Comptes Rendus de l'Académie des Sciences, Paris, vol. 73, pp. 147–154, 1871. View at Google Scholar
  49. J. F. Colombeau, Multiplication of Distributions, vol. 1532 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1992.
  50. J. F. Colombeau and A. Y. LeRoux, “Multiplications of distributions in elasticity and hydrodynamics,” Journal of Mathematical Physics, vol. 29, no. 2, pp. 315–319, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  51. C.-J. Huang, M. L. Shen, and H. H. Chang, “Propagation of a solitary wave over rigid porous beds,” Ocean Engineering, vol. 35, no. 11-12, pp. 1194–1202, 2008. View at Publisher · View at Google Scholar · View at Scopus
  52. J. Sander and K. Hutter, “On the development of the theory of the solitary wave. A historical essay,” Acta Mechanica, vol. 86, no. 1-4, pp. 111–152, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  53. J. S. Russell, “On waves,” Report of the 14th Meeting of the British Association for the Advancement of Science, 1845. View at Google Scholar
  54. H. Lamb, Hydrodynamics, Dover, 1945.
  55. J. Garnier, J. C. Muñoz Grajales, and A. Nachbin, “Effective behavior of solitary waves over random topography,” Multiscale Modeling & Simulation, vol. 6, no. 3, pp. 995–1025, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  56. G. G. Stokes, “The outskirts of the solitary wave,” Mathematical and Physical Papers, no. 5, p. 63, 1905. View at Google Scholar
  57. S.-J. Liang, J.-H. Tang, and M.-S. Wu, “Solution of shallow-water equations using least-squares finite-element method,” Acta Mechanica Sinica, vol. 24, no. 5, pp. 523–532, 2008. View at Publisher · View at Google Scholar
  58. E. V. Laitone, “The second approximation to cnoidal and solitary waves,” Journal of Fluid Mechanics, vol. 9, pp. 430–444, 1960. View at Publisher · View at Google Scholar