Step Soliton Generalized Solutions of the Shallow Water Equations

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 [2–5]. 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 [7–10]. Many other families of dispersive wave equations have been proposed as well [11–13]. Other studies attempt to include the effect of different types of bottom shape [4, 14–19]. 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 [23–25]. 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 p − p_o = ρg(h_o + η) holds; here, p is the pressure, h_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 [30–35]. 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 [37–41]. 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, $$ for τ < −τ₁, $$ for −τ₁ < τ < τ₁ and $$ 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.

E_s(Ω) and E_M(Ω) are algebras, and N(Ω) is an ideal of E_s(Ω).

The elements G of ℑ_s(Ω) are denoted by G = R₁(ϵ, x) + N(Ω). Distribution of compact support on ℝ can be embedded on ℑ_s(ℝ) by convolution with a mollifier ρ_ϵ, defined as follows: let ρ ∈ S(ℝ) (Schwartz’s space) with the properties ∫ ρ(x)dx = 1, ∫ x^αρ(x)dx = 0,    for all α ∈ N²,    | α | > 1, then we set ρ_ϵ(x): = (1/ϵ) ²ρ(x/ϵ). Then the generalized function ∫ x^αρ_ɛ(x − y)w(x)dx + N(Ω) belongs to ℑ_s(ℝ) [28].

ℑ_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 ℑ_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].

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 ς_ϵ = λcos  (ϵ(x − ct)), whereas in the solitary waves limit as ς_ϵ = λ sech (ϵ(x − ct)). 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 association ~ is stable by differentiation but not by multiplication, that is, if G₁, G₂, G ∈ ℑ_s(ℝ), and G₁ ~ G₂ then $$, but GG₁ and GG₂ are not necessarily associated.

The Heaviside generalized functions are associated between them. Moreover, Hⁿ ~ H for n ∈ ℕ, n > 0.

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

For instance, R₁(ϵ, x) = 0 if x + τ₁ ≤ 0, R₁(ϵ, x) = 1 if x + τ₁ ≥ ɛ, and R₁(ϵ, x) > 0 if 0 < x + τ₁ɛ (see Figure 1(a)). Besides, R₁(ϵ, x) = 0 if x − τ₁ ≤ 0, R₁(ϵ, x) = 1 if x − τ₁ ≥ ɛ, and R₁(ϵ, x) > 0 if 0 < x − τ₁ɛ (see Figure 1(b)). In Figure 1(c), the graph of R₁(ϵ, x + τ₁) − R₁(ϵ, x − τ₁) is shown.

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

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.

The following propositions are useful.

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:

$$ (4.1)

$$ (4.2)

where h 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 h = h_o + η, where h_o is the bottom depth, and η is the surface elevation relative to the fixed depth h_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.

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 $$ the derivative of the step soliton generalized function $$. 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 $$ where $$ 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 by

$$ (4.3a)

$$ (4.3b)

for a given τ₁, where λ and u_o are constants representing the amplitude of surface elevation and particle velocity, respectively, and h = h_o + η, where h_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 by

$$ (4.4a)

$$ (4.4b)

Proof. Using that h = h_o + η and substituting (4.3a) and (4.3b) in (4.1) with ξ = x − X(t), we obtain

$$ (4.5)

Now, using that $$, we have

$$ (4.6)

Finally, from the fact that $$, we deduce that

$$ (4.7)

Since $$ is not associate to null generalized function, such above equation implies that

$$ (4.8a)

$$ (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,

$$ (4.9)

where K is a constant. As a consequence, the soliton velocity is given by

$$ (4.10)

Now, substituting (4.3a) and (4.3b) in (4.2) and using again the fact that $$, we obtain

$$ (4.11)

or equivalently,

$$ (4.12)

Since $$ is not associate to null generalized function, from (4.12), we obtain

$$ (4.13)

Substituting (4.10) in (4.13), we have

$$ (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

$$ (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, u_o, 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/h_o, μ = (hk) 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:

$$ (4.16)

Equation (4.16) for small nonlinear parameter σ ≪ 1 holds,

$$ (4.17)

Formula (4.17) is similar to those obtained in [6, 53–55] 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, $$, which for small σ holds as follow,

$$ (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)).

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):

$$ (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]:

$$ (4.20)

where h is the height of water, u is the velocity, g is the gravity constant, and α = (1/2)(z_α/h) ² + (z_α/h) at reference depth z_α. We assume here that the bottom is constant, that is, h = h_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

$$ (4.21)

for a given τ₁, where λ and u_o are constants representing the amplitude of surface elevation and particle velocity, respectively, and h = h_o + η, where h_o is a fixed real number. Here, k, ω are the wave number and frequency, respectively. Then the following equalities hold:

$$ (4.22)

$$ (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 $$ and $$, ξ = kx − ωt and after several operations, we obtain

$$ (4.24)

Finally, taking a representant R(ϵ, ξ) = a₁ + a₂ϵξ + a₃ϵ²ξ² + a₄ϵ³ξ³ + O(ϵ⁴ξ⁴) of $$ and using the Definition 2.2, we obtain that there exist constants ν₁, ν₂ such that

$$ (4.25)

$$ (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

$$ (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 h_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)).

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.

6. Numerical Calculation of the Generalized Solution

In this section, we show a numerical procedure to find the unknown parameters λ₂ and u_o2 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.

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 $$. It is possible to check that the positive root of the above equation produces a wave amplitude with reasonable value.

We take the example described in [40] which considered the discontinuity bottom as h₁ = 80 cm, h₂ = 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/h₂, that is, M = M(x/h₂). 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/h₂) 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).

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.