Solitary wave solutions of several nonlinear PDEs modeling shallow ...

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arXiv:1709.05320v1 [nlin.SI] 15 Sep 2017

Solitary wave solutions of several nonlinear PDEs modeling shallow water waves Nikolay K. Vitanov, Tsvetelina I. Ivanova Institute of Mechanics, Bulgarian Academy of Sciences, Akad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria Abstract We apply the version of the method of simplest equation called modified method of simplest equation for obtaining exact traveling wave solutions of a class of equations that contain as particular case a nonlinear PDE that models shallow water waves in viscous fluid (Topper-Kawahara equation). As simplest equation we use a version of the Riccati equation. We obtain two exact traveling wave solutions of equations from the studied class of equations and discuss the question of imposing boundary conditions on one of these solutions.

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Introduction

Nonlinear phenomena are of large interest for modern natural and social sciences [1] - [14]. And many nonlinear phenomena are modelled by nonlinear differential equations [15] - [24]. Today the research on nonlinear phenomena is well established research area and the research on nonlinear waves finds its significant place within this area. One of the most studied topics in the research on nonlinear phenomena are the nonlinear waves and especially the water waves [25] -[32]. The goal of obtaining exact solution of model equation for water waves leaded to development of the modified method of simplest equation that will be applied below in the text [33]-[35]. Often the model equations for the nonlinear phenomena are nonlinear partial differential equations that have traveling-wave solutions. These traveling wave solutions are studied very intensively [36]-[41]. Well established methods exist for obtaining exact traveling-wave solutions of integrable nonlinear partial differential equations , e.g., the method of inverse scattering transform or the method

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of Hirota [42] - [44]. Many other approaches for obtaining exact special solutions of nonlintegrable nonlinear PDEs have been developed in the recent years (for examples see [45] - [47]). Below we shall consider the method of simplest equation and our focus will be on a version of this method called modified method of simplest equation. The method of simplest equation is based on a procedure analogous to the first step of the test for the Painleve property [48], [49]. In the version of the method called modified method of simplest equation [33] - [35] this procedure is substituted by the concept for the balance equations. The modified method of simplest equation has already numerous applications, e.g., obtaining exact traveling wave solutions of generalized Kuramoto - Sivashinsky equation, reaction - diffusion equation, reaction - telegraph equation[33], [50], generalized Swift - Hohenberg equation and generalized Rayleigh equation [34], generalized Fisher equation, generalized Huxley equation [51], generalized Degasperis - Procesi equation and b-equation [52], extended Korteweg-de Vries equation [53], etc. [54] [57]. The text below is organized as follows. In Sect. 2 we describe the methodology and formulate the problem. In Sect. 3 we present solutions of the studied nonlinear PDEs. Discussion and concluding remarks are given in Sect.4

2

Methodology and problem formulation

We shall use the part of the methodology of the modified method of simplest equation that is appropriate for solving model nonlinear PDEs for shallow water waves. This methodology is described, e.g., in [56]. The methodology works as follows. First of all by means of an appropriate ansatz the solved nonlinear partial differential equation is reduced to a nonlinear differential equation, containing derivatives of a function (1)

P (u(ξ), uξ , uξξ , . . . , ) = 0

Then the function u(ξ) is searched as some function of another function, e.g., g(ξ), etc., i.e. (2)

u(ξ) = G[g(ξ)]

The kind of the function F is not prescribed. Often one uses a finite-series relationship, e.g., (3)

u(ξ) =

ν2 X

µ1 =−ν1

2

pµ1 [g(ξ)]µ1 ;

where pµ1 are coefficients (such relationship will be used below too). The function g(ξ) is solution of simpler ordinary differential equation called simplest equation. Eq.(2) is substituted in Eq.(1) and let the result of this substitution be a polynomial containing g(ξ). Next a balance procedure is applied. This procedure has to ensure that all of the coefficients of the obtained polynomial of g(ξ) contain more than one term. The procedure leads to a balance equation for some of the parameters of the solved equation and for some of the parameters of the solution. Eq.(2) describes a candidate for solution of Eq.(1) if all coefficients of the obtained polynomial of are equal to 0. This condition leads to a system of nonlinear algebraic equations for the coefficients of the solved nonlinear PDE and for the coefficients of the solution. Any nontrivial solution of this algebraic system leads to a solution the studied nonlinear partial differential equation. We shall apply the above methodology to the class of equations (4)

∂u ∂u ∂2u ∂3u ∂um +u + r 2 + δ 3 + σun m = 0 ∂t ∂x ∂x ∂x ∂x

where σ, r and δ are parameters and n and m are non-negative natural numbers. For the case n = 0, m = 4, Eq.(4) is reduced to an equation modeling long nonlinear waves in viscous fluid (Topper - Kawahara equation) [58]. We shall search a traveling wave solution of Eq.(4) in the form (5)

u(ξ) =

ν X

pµ g(ξ)µ

µ=0

where ν and pµ are parameters and ξ = αx + βt. The simplest equation for g(ξ) will be Θ

g′ =

(6)

dg X θ = qθ g dξ θ=0

where qθ is parameter. Next step is to obtain the possible balance equations by balancing the powers of g arising from the terms from Eq.(4). The maximum powers of g arising from the 5 terms in Eq.(4) are ∂u ∂u → ν + Θ − 1; Term u → 2ν + Θ − 1; ∂t ∂x ∂2u ∂3u Term r 2 → ν + 2(Θ − 1); Term δ 3 → ν + 3(Θ − 1); ∂x ∂x m ∂u Term αun m → nν + m(Θ − 1) ∂x Term

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Thus the possible members of a balance equation are A = 2ν + (Θ − 1); B = ν + 3(Θ − 1); C = (n + 1)ν + m(Θ − 1) Let us consider the possible balance equations from the point of view of increasing values of the parameter m. We have the following possibilities 1. m = 1. The interesting cases are n = 0 or n ≥ 2. (a) n = 0. The balance equation is A = B. Thus ν = 2(Θ − 1).  (b) n ≥ 2. The balance equation is B = C. Thus ν = 2 Θ−1 . n

2. m = 2. The interesting cases are: n = 1 and n > 1

(a) n = 1. The balance equation is B = C. Thus ν = (Θ − 1).  (b) n > 1. The balance equation is B = C. Thus ν = Θ−1 . n

3. m ≥ 3. The interesting case is n ≥ 0.

(a) n ≥ 0. The balance equation is A = C. Thus ν =

m−1 1−n



(Θ − 1).

Below we shall consider examples from the case m = 4 and m = 2.

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Several solutions of equations of class (4)

Let us first discuss the case m = 4. As we can see from the above list we must have n = 0 and then the balance equation becomes ν = 3(Θ − 1). Let us discuss the case Θ = 2. Then ν = 3. All above means that we discuss the equation (7)

∂u ∂u ∂2u ∂3u ∂u4 +u +r 2 +δ 3 +σ 4 =0 ∂t ∂x ∂x ∂x ∂x

and search for the solution of the kind (8)

u(ξ) = p0 + p1 g(ξ) + p2 g(ξ)2 + p3 g(ξ)3

where the simplest equation for g(ξ) is (9)

g ′ = q0 + q1 g + q2 g 2

We substitute Eqs. (8) and (9) in Eq.(7) and obtain the following equation (10) A0 + A1 g(ξ) + A2 g(ξ)2 + A3 g(ξ)3 + A4 g(ξ)4 + A5 g(ξ)5 + A6 g(ξ)6 + A7 g(ξ)7 = 0 4

where Ai , i = 0, . . . , 7, are nonlinear algebraic relationship between parameters of the equation and parameters of the solution. Further we set Ai = 0 and obtain, e.g., the following solution of the system of 8 nonlinear algebraic equations σ = q1 = p3 = p2 = p1 = p0 =

δ2 16r (q0 q2 α2 δ 2 + 4r 2 )1/2 2 αδ 2 2 3 15 δ α q2 − 2 r   15 2 3 2 2 2 1/2 − αq2 δ (q0 q2 α δ + 4r ) + 2 2 r # " 15 q2 − 3q0 q2 α2 δ 2 + 8r 2 + 4r(q0 q2 α2 δ 2 + 4r 2 )1/2 2 r " # 2 δ 1 2βδ + 30q0 q2 α2 δ 2 + 48r 2 + 15q0 q2 (q0 q2 α2 δ 2 + 4r 2 )1/2 − 2αδ r

(11) Eq.(9) is a particular case of equation of Riccati. One particular solution of the Riccati equation is hπ i π q1 − tanh (ξ + ξ0 ) (12) g(ξ) = − 2q2 2q2 2 where ξ0 is a constant of integration and π has to satisfy π 2 = q12 − 4q0 q2 > 0. The substitution of q1 from Eq.(11) in the last inequality leads to q12 = 16r 2 /(α2 δ 2 ) which is larger than 0 for nonzero values of the real parameters r and δ. Let us set π = 4r/(αδ). Then the solution (12) of Eq.(11) becomes     1 2r 2 2 2 1/2 (13) g(ξ) = − q0 q2 α δ + 4r + 2r tanh (ξ + ξ0 ) q2 αδ αδ

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The substitution of Eq.(13) in Eq.(8) leads to the following solution of Eq.(7) " # 2 1 δ u(ξ) = − 2βδ + 30q0 q2 α2 δ 2 + 48r 2 + 15q0 q2 (q0 q2 α2 δ 2 + 4r 2)1/2 + 2αδ r # " 15 3q0 q2 α2 δ 2 + 8r 2 + 4r(q0 q2 α2 δ 2 + 4r 2 )1/2 × 2αδr     2r 2 2 2 1/2 q0 q2 α δ + 4r + 2r tanh (ξ + ξ0 ) + αδ   2   2r 15q2 3 2 2 2 1/2 2 2 2 1/2 (q0 q2 α δ + 4r ) + 2 q0 q2 α δ + 4r (ξ + ξ0 ) + 2r tanh + 2 r αδ   3  2r 15αδq22 2 2 2 1/2 q0 q2 α δ + 4r + 2r tanh (ξ + ξ0 ) (14) 2r αδ

Let us now consider another example for solution of equation of the class (4). In this example we set m = 2 and n = 1 in Eq.(4). We shall assume Θ = 2 and thus from the balance equation ν = Θ − 1 we obtain ν = 1. Then we shall search solution of the kind (15)

u(ξ) = p0 + p1 g(ξ)

where the differential equation of g(ξ) is again the Riccati equation (9). The substitution of Eqs. (15)and (9) in Eq.(4) (where m = 2 and n = 1) leads to the following equation (16)

A0 + A1 g(ξ) + A2 g(ξ)2 + A3 g(ξ)3 + A4 g(ξ)4 = 0

Thus we obtain the following 5 nonlinear algebraic equations for the parameters of the equation and parameters of the solution A0 = 3δq2 + σp1 = 0 A1 = 12δαq1 q2 + p1 + 2rq2 + 2σαp0 q2 + 3σαp1q1 = 0 A2 = βq2 + αp0 q2 + αp1 q1 + 3rαq1q2 + 8δα2 q0 q22 + 7δα2 q12 q2 + 3σα2 p0 q1 q2 + 2σα2 p1 q0 q2 + σα2 p1 q12 = 0 A3 = βq1 + αp0 q1 + αp1 q0 + 2rαq0q2 + rαq12 + 2σα2 p0 q0 q2 + σα2 p0 q12 + σα2 p1 q0 q1 + 8δα2 q0 q1 q2 + δα2 q13 = 0 A4 = β + σα2 p0 q1 + αp0 + 2δα2q0 q2 + δα2 q12 + rαq1 = 0 (17) 6

One solution of this system of equations is σp1 3δ 2σ 2 αp0 + 2rσ − 3δ = − 3αδσ 4 2 2 2σ α p0 + 4rσ 3 αp0 + 2r 2 σ 2 + 3δrσ − 9δ 2 − 6αp0 σ 2 δ − 9βσ 2 δ = − 6p1 σ 3 δα2

q2 = − q1 q0 (18)

We shall use the solution (12) of the equation of Riccati (9). In this case we must have 2rσ − 2βσ 2 − 3δ π= >0 δα2 σ 2 When this relationship is fulfilled then the solution of the simplest equation (9) is   2rσ − 2βσ 2 − 3δ 2σ 2 αp0 + 2rσ − 3δ 3(3δ − 2rσ + 2βσ) − tanh (ξ + ξ0 ) g(ξ) = − 2ασ 2 p1 2α2 σ 3 p1 2δα2 σ 2 (19) The solution of the solved equation from class (4) (m = 2, n = 1) is (  ) 2rσ − 2βσ 2 − 3δ 2σ 2 αp0 + 2rσ − 3δ 3(3δ − 2rσ + 2βσ) + tanh (ξ + ξ0 ) u(ξ) = p0 − p1 2ασ 2 p1 2α2 σ 3 p1 2δα2 σ 2 (20)

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Discussion and concluding remarks

The study above has demonstrated the capability of the modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations connected to water waves theory. Assuming vary large size of the system along the x-axis of the co-ordinate system we can impose boundary conditions that can further fix the parameters of the solutions. Let us demonstrate this on the basis of solution (20). We can impose, e.g. the following boundary conditions (21)

u(ξ = ∞) = d1 ; u(ξ = −∞) = d2

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where d1 and d2 are some appropriate constants. On the basis of Eq.(20) we arrive at the system of equations

(22)

2σ 2 αp0 + 2rσ − 3δ 3(δ − 2rσ + 2βσ) p0 − + = d1 2ασ 2 2α2 σ 3 2σ 2 αp0 + 2rσ − 3δ 3(δ − 2rσ + 2βσ) p0 − − = d2 2ασ 2 2α2 σ 3

From Eq.(22) we can determine two of the parameters, e.g, α and β. We obtain α =

3δ − 2rσ σ 2 [2p0 + p1 (d1 + d2 )]

" 1 β = 9δ 2 p1 (d1 − d2 ) − 36δσp20 + 4r 2 σ 2 p1 (d1 − d2 ) − 3 2 6σ [2p0 + p1 (d1 + d2 )] 12δrσ(d1 − d2 )p1 + 24rσ 2p20 + 12rσ 2d1 p21 d2 + 24rσ 2 p0 (d1 + d2 )p1 − 18δσd1p21 d2 − # 36δσp0 (d2 − d1 )p1 + 6rσ 2 (d21 + d22 )p21 − 9δσ(d21 + d22 )p21 (23) Thus the solution (20) becomes ( ( 2rσ − 2βσ 2 − 3δ 2σ 2 αp0 + 2rσ − 3δ 3(3δ − 2rσ + 2βσ) + tanh × u(ξ) = p0 − p1 2ασ 2 p1 2α2 σ 3 p1 2δα2σ 2 " " 3δ − 2rσ 1 x+ 3 9δ 2 p1 (d1 − d2 ) − 2 2 σ [2p0 + p1 (d1 + d2 )] 6σ [2p0 + p1 (d1 + d2 )] 36δσp20 + 4r 2 σ 2 p1 (d1 − d2 ) − 12δrσ(d1 − d2 )p1 + 24rσ 2 p20 + 12rσ 2 d1 p21 d2 + 24rσ 2 p0 (d1 + d2 )p1 − 18δσd1p21 d2 − 36δσp0(d2 − d1 )p1 + #)) #

6rσ 2 (d21 + d22 )p21 − 9δσ(d21 + d22 )p21 t + ξ0 (24)

This solution has many free parameters. This means that additional boundary conditions can be imposed, e.g. u′ (x = −∞) = 0; u′ (∞) = 0, etc. We note that the solutions obtained above are traveling waves of kink kind. One may search also for solitary waves or for periodic waves. The results of this research will be reported elsewhere.

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