Advanced Studies in Theoretical PhysicsVol. 11, 2017, no. 12, 593 - 599

HIKARI Ltd, www.m-hikari.comhttps://doi.org/10.12988/astp.2017.7944

Soliton Solutions for a Generalized

Shallow Water Model

Cesar A. Gomez

Department of MathematicsUniversidad Nacional de Colombia, Bogota Colombia

Copyright c© 2017 Cesar A. Gomez. This article is distributed under the Creative Com-

mons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Abstract

We construct exact solutions for a new model, which can be con-sidered as a generalization of the Whitham-Broer-Kaup equation. Thenew model have variable coefficients (depending of the temporal vari-able) and forcing terms. We use the tanh-coth method for obtain exactsolutions for this model, from which, solutions for the classic Whitham-Broer-Kaup equation (WBK) can be derived as particular cases. Wemade the graphs of some of the solutions with the aim to compare thesolutions in the case of variable coefficients with those that have con-stant coefficients and without forcing terms.

Subject Classification: 35C05

Keywords: Improved tanh-coth method; soliton solutions; variable coef-ficients; Witham-Broer-Kaup equation; forcing term

1 Introduction

In the study of nonlinear partial differential equations (NPDE’s) or systems ofNLPDE’s, the generalized models have great relevance today in the sense thatwe can obtain the results on the classics models and furthermore, we can toobserve new structures respect to the physical phenomena that them describe.On the other hand, the mathematical software developed in the last decades

594 Cesar A. Gomez

helped to the investigators to implement computational methods to obtainexact solutions to several important models. Some of the more used are: theExp.(-φ(ξ)) method [1], the tanh-coth method [2], and the improved tanh-cothmethod [3], however, others models can be studied using some analytical toolssuch as: Hirota method [4], Lie groups [5]. Clearly, the solutions obtainedby means of computational methods are used by the researches with the aimto compare analytical results, to compare numerical results and furthermore,help us to have a better approach to the phenomena that the model describe.The main purpose of this work, is to obtain exact solutions for the followingnew model (generalization of the Whitham-Broer-Kaup equation [6])

{ut(x, t) + δ(t)u(x, t)ux(x, t) + ρ(t)vx(x, t) + µ(t)uxx(x, t) = F (t),

vt(x, t) + σ(t)[v(x, t)u(x, t)]x + θ(t)uxxx(x, t) + ω(t)vxx(x, t) = G(t).(1)

The use of variable coefficients (depending on the temporal variable) is in-teresting in the sense that with this, the model can be handled as a morerealistic approach to the phenomena that the model describe, several and newstructures of the solutions can be seen. See for instance the references [7][8][9]and references therein, where the authors haves solve NLPDE’s with vari-able coefficients. With respect to (1), if we take δ(t) = ρ(t) = σ(t) = 1,µ(t) = a = −ω(t), θ(t) = b, con a, b constants and F (t) = G(t) = 0, we obtainthe following classical Whitham-Broer-Kaup equation [6]{

ut(x, t) + u(x, t)ux(x, t) + vx(x, t) + auxx(x, t) = 0,

vt(x, t) + [v(x, t)u(x, t)]x + buxxx(x, t)− avxx(x, t) = 0.(2)

Details and exact solutions for (2) can be seen in [6]. However, one of the ob-jective of our work is to show that, solutions to (2) can be derived as particularcase of the solutions derived for (1). With this end in mint, we will use theimproved tanh-coth method wit the aim to obtain exact solutions for (1). Thepaper is organizes as follows: In Sec. 2, we obtain exact solutions for (1) usingthe improved tanh-coth method and as a consequence, we can derive solutionsfor (2). In Sec. 3, we made the graphs of some solutions to compare its withthe solutions of the classical WBK equation (2). Finally, some conclusions aregiven.

2 searching solutions for (1)

In this section, we will use the technique given by the improved tanh-cothmethod [3]. With this end, we consider the transformation

Soliton solutions for a generalized shallow water model 595

u(x, t) = u(ξ) +

∫F (t)dt,

v(x, t) = v(ξ) +∫G(t)dt,

ξ = x+ λt+ ξ0,

(3)

where for sake of simplicity, we use the same letters u, v. Substituting (3) into(1) we have the following system of ordinary differential equations

λ(t)u′(ξ) + δ(t)u(ξ)u′(ξ) + δ(t)(

∫Fdt)u′(ξ) + ρ(t)v′(ξ) + µ(t)u′′(ξ) = 0,

λ(t)v′(ξ) + σ(t)v′(ξ)u(ξ) + σ(t)(∫F (t)dt)v′(ξ) + σ(t)v(ξ)u′(ξ)+

σ(t)(∫G(t)dt)u′(ξ) + θ(t)u′′′(ξ) + ω(t)v′′(ξ) = 0.

(4)Here ”′” denote the ordinary derivation respect to ξ, u′(ξ) = du

dξ. Next, we

consider solutions for (4) in the formu(ξ) =

∑Mi=0 ai(t)φ(ξ)i +

∑2Mi=M+1 ai(t)φ(ξ)M−i,

w(ξ) =∑N

i=0 bi(t)φ(ξ)i +∑2N

i=N+1 bi(t)φ(ξ)N−i,

(5)

Where φ(ξ) satisfies the Riccati equation (10)

φ′(ξ) = γ(t)φ2(ξ) + β(t)φ(ξ) + α(t), (6)

where α(t), β(t) and γ(t) are functions to determinate later. Balancing u′′

with uu′ in the first equation of (5), we have M + 2 = M + M + 1 so thatM = 1. In the same form, balancing u′′′ with v′u in the second equation of(5), we have M + 3 = M +N + 1 and N = 2. Therefore, (5) take the form

{u(ξ) = a0(t) + a1(t)φ(ξ) + a2(t)φ(ξ)−1,

v(ξ) = b0(t) + b1(t)φ(ξ) + b2(t)φ(ξ)2 + b3(t)φ(ξ)−1 + b4(t)φ(ξ)−2.(7)

Substitution of (7) into (4) and taking into account (6) lead us to a system ofalgebraic equations in the unknowns α(t), β(t), γ(t), λ(t), a0(t), a1(t), a2(t),b0(t), b1(t), b2(t), b3(t), b4(t) which by reasons of space we omit here. Solvingthis system with aid of the Mathematica software we obtain a lot of solutions,however, for sake of simplicity and with the aim to illustrate the results, weconsider only the following, which give us the more general solution

596 Cesar A. Gomez

a0(t) = 12δ(t)γ(t)(δ(t)−σ(t))

[− 2δ(t)2γ(t)(

∫F (t)dt)+

3β(t)(√

γ2(t) (4δ(t)ρ(t)σ(t)θ(t) + (µ(t)σ(t)− δ(t)ω(t))2) + γ(t)µ(t)σ(t))

+

δ(t)γ(t)(β(t)(ω(t)− 2µ(t)) + 2σ(t)(∫F (t)dt))

],

a1(t) = − 12δ(t)σ(t)[√

(2δ(t)γ(t)ω(t) + 2γ(t)µ(t)σ(t))2 − 4δ(t)σ(t) (4γ2(t)µ(t)ω(t)− 4ρ(t)γ2(t)θ(t))

+2δ(t)γ(t)ω(t) + 2γ(t)µ(t)σ(t)], a2(t) = 0,

b0(t) = − 1ρ(t)σ(t)2[

α(t)ω(t)√γ2(t) (4δ(t)ρ(t)σ(t)θ(t) + (µ(t)σ(t)− δ(t)ω(t))2)+

α(t)ω(t) (δ(t)γ(t)ω(t) + γ(t)(−µ(t))σ(t)) +

ρ(t)σ(t)(σ(t)(

∫G(t)dt) + θ(t) (2α(t)γ(t) + β2(t))

) ],

b2(t) = − 1ρ(t)σ(t)2

[γ(t)ω(t)

√γ2(t) (4δ(t)ρ(t)σ(t)θ(t) + (µ(t)σ(t)− δ(t)ω(t))2)+

γ(t) (δ(t)γ(t)ω(t)2 + 2ρ(t)γ(t)σ(t)θ(t)− γ(t)µ(t)σ(t)ω(t))],

b1(t) = b3(t) = b4(t) = 0,

λ(t) = − 12γ(t)(δ(t)−σ(t))[

3β(t)√γ2(t) (4δ(t)ρ(t)σ(t)θ(t) + (µ(t)σ(t)− δ(t)ω(t))2)+

β(t) (δ(t)γ(t)ω(t) + γ(t)µ(t)σ(t))]

(8)With this set of values, the respective solution to (6) is given by (see 10)

φ(ξ) =−√β2(t)− 4α(t)γ(t) tanh[1

2

√β2(t)− 4α(t)γ(t)(ξ + ξ0)]− β(t)

2γ(t), (9)

where α(t), β(t) and γ(t) are arbitrary functions depending on t. Accordingwith (3) and (7) we have the following solutions for (1){

u(x, t) = a0(t) + a1(t)φ(ξ) +∫F (t)dt,

v(x, t) = b0(t) + b2(t)φ(ξ)2 +∫G(t)dt,

(10)

where φ(ξ) is given by (9), ξ = x+λ(t) t+ ξ0, a0(t), a2(t), b0(t), b2(t) and λ(t)the values that appear in (8), ξ0 arbitrary constant.

Soliton solutions for a generalized shallow water model 597

3 Results and Discussion

The following are the graph of some of the obtained solutions for x ∈ [0, 1000]and t ∈ [0, 10],

(a) u1 (b) u2 (c) u3

Figure 1: u(x, t).

(a) v1 (b) v2 (c) v3

Figure 2: v(x, t)

For u1, v1 we obtain the graph of the system (2) (constant coefficients andwithout forcing term) where we have used the following values: α(t) = 1,β(t) = 3, γ(t) = 1, δ(t) = 2, ρ(t) = 1, µ(t) = 5, σ(t) = 1, θ(t) = 1, ω(t) = −5,F (t) = 0, G(t) = 0. On the other hand, u2, v2 are the graph correspondent to(1) with constant coefficients and forcing term. In this case, we have used thesame values of the previous, but F (t) = cos t and G(t) = 3 cos t. Finally, foru3, v3 we have used the following values: α(t) = 1, β(t) = 3, γ(t) = 1, δ(t) = 2,ρ(t) = 1, µ(t) = 5t2, σ(t) = 1, θ(t) = 1, ω(t) = −5t2, F (t) = cos t, G(t) =3 cos t. In general, the graph corresponding to (2) have the same structure,independent of the values of the coefficients. Clearly, taking into account theseveral solutions (periodic or type soliton) depending of the solutions for (6)according with the values of its coefficients (see [10]). However, as can be seenfrom the graphs, when a forcing term is used and variable coefficients are used,the solutions taken several forms (see for instance the graph of u2, v2,u3, v3),with structures very different to those in the case of constant coefficients andwithout forcing term. This fact can be used to have a best approach of thephenomena modeled by the WBK equation.

598 Cesar A. Gomez

4 Conclusion

We have obtained exact solutions for a new generalized model, which includeas particular case the Whitham-Broer-Kaup equation. From the obtainedsolutions we have derived and draw the respective solutions for the classicalWBK equation (constant coefficients and without forcing term). Clearly, thegeneralized model give us many different structures of the possible form ofthe solutions, when the model have a forcing term and variable coefficients,which can be give us a best understanding of the phenomena modeled by theWBK equation. The results can be help to researches that are work in theconstruction of building near to the sea.

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Received: October 10, 2017; Published: October 29, 2017