Traveling wave solutions of the generalized scale-invariant analog of the KdV equation by tanh–coth method

Oswaldo González-Gaxiola; Juan Ruiz de Chávez

doi:10.1515/nleng-2022-0325

Article Open Access

Traveling wave solutions of the generalized scale-invariant analog of the KdV equation by tanh–coth method

Oswaldo González-Gaxiola and Juan Ruiz de Chávez

Published/Copyright: October 3, 2023

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From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

In this work, the generalized scale-invariant analog of the Korteweg–de Vries equation is studied. For the first time, the tanh–coth methodology is used to find traveling wave solutions for this nonlinear equation. The considered generalized equation is a connection between the well-known Korteweg–de Vries (KdV) equation and the recently investigated scale-invariant of the dependent variable (SIdV) equation. The obtained results show many families of solutions for the model, indicating that this equation also shares bell-shaped solutions with KdV and SIdV, as previously documented by other researchers. Finally, by executing the symbolic computation, we demonstrate that the used technique is a valuable and effective mathematical tool that can be used to solve problems that arise in the cross-disciplinary nonlinear sciences.

Keywords: KdV equation; SIdV equation; the tanh–coth method; traveling waves; symbolic computation

MSC 2010: 35C05; 35C07; 35Q53; 68W30

1 Introduction

Many fields of science and engineering depend heavily on the mathematical models presented by nonlinear partial differential equations (NPDEs) to explain complex phenomena. These fields include electromagnetic wave theory, ocean dynamics, plasma physics, fluid mechanics, field theory, nonlinear optical fibers, nuclear physics, ion acoustic waves, biological process engineering, chemical kinetics, climatological phenomena, and several other mathematical physics problems. In 1895, Dutch mathematicians D. J. Korteweg and G. de Vries developed the Korteweg–de Vries (KdV) equation in a formal manner, and the KdV equation is an NPDE that models the propagation of long waves on shallow water, and this simultaneously includes weak advective nonlinearity and dispersion effects given by Korteweg et al. [1]

(1) u t + 6 u u x + u x x x = 0 ,

where u is the perturbation wave function that depends on the spatial variable x and time t . It is well known that Eq. (1) has bell-shaped solutions of the type:

(2) u ( x , t ) = c 2 sech 2 c 2 ( x − c t ) ,

where c is the velocity of the wavefront [2].

Despite its antiquity, the KdV equation is still an active area of study and research, with several articles published on the topic in recent years. For instance, Sen et al. [3] investigated the KdV equation and derived generalizations using a one-parametric family of advective velocities. Triki et al. [4] studied the KdV equation of the fifth order with free coefficients. The KdV equation’s border control problem is examined in detail the study by Liang et al. [5]. A novel integrable nonlocal-modified KdV equation is discussed in the study by Wazwaz [6]. Several extensions and adaptations of the KdV equation are analyzed by Wazwaz [7]. For a nonholonomic modification of the KdV equation, Mirzazadeh et al. [8] obtained 1-solitons. An algebraic approach is used to investigate an extension of the KdV equation in the study by Gonzalez-Gaxiola et al. [9]. Exact solutions for the KdV equation with a source term are found in previous studies [10,11]; a generalized KdV equation with time-dependent attenuation and dispersion is explored in the study by Biswas [12]; novel studies on the KdV equation and some generalizations taking into account different physical effects as well as fractional derivatives can be found in previous studies [13–21]; and these are just a few examples.

Among the numerous studies associated with the KdV equation that have been published in the last few decades, several studies propose to generalize and/or modify the KdV equation. In 2012, for instance, Sen et al. [3] presented the modified KdV equation:

(3) u t + 2 u x x u u x = u x x x .

Eq. (3) is invariant under scaling of the dependent variable; and is therefore referred to as SIdV; here, Eq. (3) was discovered by surprise using computer approaches when researchers explored equations with bell-shaped solutions analogous to the KdV equation. Other studies on Eq. (3) have been published and can be found in previous studies [22–24].

In this article, we will consider the generalized scale-invariant analog of the Korteweg–de Vries (gsiaKdV) equation [25], which is an NPDE and whose dimensionless form is given by,

(4) u t + 3 ( 1 − α ) u + ( 1 + α ) u x x u u x = γ u x x x ,

where u x x x is a dispersion term, while the term ( 3 ( 1 − α ) u + ( 1 + α ) u x x u ) can be seen as an advecting velocity. First, let us observe that, if α = − 1 and γ = − 1 , then Eq. (4) reduces to the well-known KdV Eq. (1). Second, we can observe that if α = 1 and γ = 1 then Eq. (4) reduces to SIdV Eq. (3).

Eq. (4) was investigated in the study by Fan et al. [26], and the authors demonstrated the existence of traveling waves of the bell and valley types. Using the tanh–coth method for the first time, the main objective of this research is to find new traveling wave-type solutions for Eq. (4) with α ≠ ± 1 and γ ≠ 0 .

In virtue of the significance of the previously described gsiaKdV equation, we will conduct a study to obtain solutions of the traveling wave type for the first time using the tanh–coth technique. In addition, some 3D and 2D propagation profiles for the derived solutions will be discussed by selecting various parameters that describe the solution sets achieved by the used strategy.

The structure of the article is the following. We present an overview of the methods used in Section 2. In Section 3, we use the proposed method to find multiple families of gsiaKdV equation solutions. Section 4 displays a graphical representation of some of the solutions generated for various parameter values. The graphical results and some KdV equation variants are briefly discussed in Section 5. Finally, in Section 6, we summarize our findings and present our final conclusions.

2 Brief description of the tanh–coth method

The tanh–coth method originally established in previous studies [27,28] provides a very useful methodology for finding traveling wave-type solutions of NPDE. We will explain how to implement the method in the rest of this section.

(I) Consider the general nonlinear partial differential equation given by:

(5) G ( u , u t , u x , u x x , u x x x , … ) = 0 .

Using traveling wave variable change u ( x , t ) = u ( ξ ) with ξ = c ( x − ω t ) , Eq. (5) becomes the ordinary differential equation:

(6) F ( u , − ω u ξ , c u ξ , c 2 u ξ ξ , c 3 u ξ ξ ξ , … ) = 0 .

(II) The tanh–coth method provides the solutions for Eq. (6) as the finite sum:

(7) u ( ξ ) = S ( Y ) = ∑ i = 0 M a i Y i ( ξ ) + ∑ i = 1 M b i Y − i ( ξ ) ,

where the coefficients a i and b i are constants to be determined and Y is a new dependent variable introduced by the method and is given by:

(8) Y = tanh ( ξ ) .

The introduction of this new dependent variable implies that:

u ξ = ( 1 − Y 2 ) d S d Y ,

(9) u ξ ξ = − 2 Y ( 1 − Y 2 ) d S d Y + ( 1 − Y 2 ) 2 d 2 S d Y 2 ,

u ξ ξ ξ = 2 ( 1 − Y 2 ) ( 3 Y 2 − 1 ) d S d Y − 6 Y ( 1 − Y 2 ) 2 d 2 S d Y 2 + ( 1 − Y 2 ) 3 d 3 S d Y 3 .

The subsequent derivatives can be computed in a similar way.

(III) To determine the upper limit M of the sum in Eq. (7), the linear terms of highest order in the resulting equation with the highest order nonlinear terms are balanced.

(IV) We consider u ( ξ ) given in Eq. (7) and the necessary derivatives u ξ , u ξ ξ , u ξ ξ ξ , … , which can be calculated as in Eq. (9), to substitute in the ordinary differential Eq. (6) and thus, we will obtain the polynomial equation:

(10) P [ Y ] = 0 .

(V) We select all the terms that have the same algebraic power of Y from the polynomial Eq. (10), we set them equal to zero and obtain a nonlinear system of algebraic equations with the set of unknown parameters { a 0 , … , a M , b 1 , … , b M , c , ω } . Using software such as Mathematica, we can execute symbolic calculations to solve the algebraic equations with the natural restrictions of the mathematical model.

(VI) Finally, having obtained the coefficients { a 0 , … , a M , b 1 , … , b M , c , ω } and considering the equality (7), one can obtain the exact solutions to Eq. (5).

3 Utilization of the tanh–coth methodology

The traveling wave transform of Eq. (4) is assumed to be of the form u ( x , t ) = U ( ξ ) , where ξ = c ( x − ω t ) . We calculate using the well-known chain rule:

(11) u t = − c ω U ξ , u x = c U ξ , u x x = c 2 U ξ ξ , and u x x x = c 3 U ξ ξ ξ .

Substituting directly into Eq. (4), we achieve the nonlinear ordinary differential equation:

(12) − c ω U U ξ + 3 c ( 1 − α ) U 2 U ξ + c 3 ( 1 + α ) U ξ U ξ ξ − γ c 3 U U ξ ξ ξ = 0 .

Integrating once with respect to ξ and considering the constants of integration as null, we obtain

(13) − c ω U 2 + 2 c ( 1 − α ) U 3 + c 3 ( 1 + α + γ ) U ξ 2 − 2 γ c 3 U U ξ ξ = 0 .

Then, using the characteristic variable change of the method, i.e., Y = tanh ( ξ ) and considering Eq. (7), the last differential equation is rewritten as:

(14) − ω c S 2 + 2 c ( 1 − α ) S 3 + c 3 ( 1 + α + γ ) ( 1 − Y 2 ) 2 d S d Y 2 − 2 γ c 3 S ( 1 − Y 2 ) 2 d 2 S d Y 2 − 2 Y ( 1 − Y 2 ) d S d Y = 0 .

Balancing S 3 with S ⋅ d 2 S d Y 2 gives M = 2 . Consequently, the tanh–coth technique enables the use of the finite sum:

(15) u ( x , t ) = U ( ξ ) = a 0 + a 1 Y + a 2 Y 2 + b 1 Y − 1 + b 2 Y − 2 .

Substituting Eq. (15) with their respective derivatives into Eq. (14) and collecting all terms with equal power of Y , after some algebraic simplification, we obtain the following nonlinear system of algebraic equations:

12 a 1 b 1 γ c 3 + 4 a 0 b 2 γ c 3 + 48 a 2 b 2 γ c 3 + 4 a 1 b 1 c 3 α + 16 a 2 b 2 c 3 α + 4 a 1 b 1 c 3 + 16 a 2 b 2 c 3 − 6 a 2 b 1 2 c α − 12 a 0 a 1 b 1 c α − 6 a 1 2 b 2 c α − 12 a 0 a 2 b 2 c α − 2 a 1 b 1 c ω − 2 a 2 b 2 c ω + 6 a 2 b 1 2 c + 12 a 0 a 1 b 1 c + 6 a 1 2 b 2 c + 12 a 0 a 2 b 2 c + a 1 2 γ c 3 − 4 a 0 a 2 γ c 3 + a 1 2 c 3 α + a 1 2 c 3 − 2 a 0 3 c α − a 0 2 c ω + 2 a 0 3 c + 5 b 1 2 γ c 3 − 8 b 2 2 γ c 3 + b 1 2 c 3 α + b 1 2 c 3 = 0 ,

− 7 a 1 2 γ c 3 + 8 a 2 2 γ c 3 − 20 a 0 a 2 γ c 3 + a 1 2 c 3 α − 8 a 2 2 c 3 α + a 1 2 c 3 − 8 a 2 2 c 3 − 6 a 0 a 2 2 c α − 6 a 1 2 a 2 c α − a 2 2 c ω + 6 a 0 a 2 2 c + 6 a 1 2 a 2 c = 0 ,

4 a 1 2 γ c 3 − 16 a 2 2 γ c 3 + 8 a 0 a 2 γ c 3 + 4 a 2 2 c 3 α + 4 a 2 2 c 3 − 2 a 2 3 c α + 2 a 2 3 γ = 0 ,

4 a 2 b 1 γ α + 4 a 0 a 1 γ c 3 − 24 a 1 a 2 γ c 3 + 4 a 1 a 2 c 3 α + 4 a 1 a 2 c 3 − 6 a 1 a 2 2 c α + 6 a 1 a 2 2 ω = 0 ,

− 4 a 0 b 1 γ c 3 − 20 a 2 b 1 γ c 3 − 4 a 1 b 2 γ c 3 − 4 a 2 b 1 c 3 α − 4 a 2 b 1 c 3 − 6 a 2 2 b 1 c α + 6 a 2 2 b 1 c − 8 a 0 a 1 γ c 3 + 12 a 1 a 2 γ c 3 − 8 a 1 a 2 c 3 α − 8 a 1 a 2 c 3 − 2 a 1 3 c α − 12 a 0 a 1 a 2 c α − 2 a 1 a 2 c ω + 2 a 1 3 c + 12 a 0 a 1 a 2 c = 0 ,

− 6 a 1 b 1 γ c 3 − 8 a 0 b 2 γ c 3 − 24 a 2 b 2 γ c 3 − 2 a 1 b 1 c 3 α − 8 a 2 b 2 c 3 α − 2 a 1 b 1 c 3 − 8 a 2 b 2 c 3 − 12 a 1 a 2 b 1 c α − 6 a 2 2 b 2 c α + 12 a 1 a 2 b 1 c + 6 a 2 2 b 2 c + 2 a 1 2 γ c 3 + 16 a 0 a 2 γ c 3 − 2 a 1 2 c 3 α + 4 a 2 2 c 3 α − 2 a 1 2 c 3 + 4 a 2 2 c 3 − 6 a 0 a 1 2 c α − 6 a 0 2 a 2 c α − a 1 2 c ω − 2 a 0 a 2 c ω + 6 a 0 a 1 2 c + 6 a 0 2 a 2 c − 4 b 1 2 γ c 3 = 0 ,

4 a 0 b 1 γ c 3 + 28 a 2 b 1 γ c 3 − 8 a 1 b 2 γ c 3 + 8 a 2 b 1 c 3 α − 4 a 1 b 2 c 3 α + 8 a 2 b 1 c 3 − 4 a 1 b 2 c 3 − 6 a 1 2 b 1 c α − 12 a 0 a 2 b 1 c α − 12 a 1 a 2 b 2 c α − 2 a 2 b 1 c ω + 6 a 1 2 b 1 c + 12 a 0 a 2 b 1 c + 12 a 1 a 2 b 2 c + 4 a 0 a 1 γ c 3 + 4 a 1 a 2 c 3 α + 4 a 1 a 2 c 3 − 6 a 0 2 a 1 c α − 2 a 0 a 1 c ω + 6 a 0 2 a 1 c − 12 b 1 b 2 γ c 3 = 0 ,

4 a 0 b 1 γ c 3 − 12 a 2 b 1 γ c 3 + 28 a 1 b 2 γ c 3 − 4 a 2 b 1 c 3 α + 8 a 1 b 2 c 3 α − 4 a 2 b 1 c 3 + 8 a 1 b 2 c 3 − 6 a 1 b 1 2 c α − 6 a 0 2 b 1 c α − 12 a 0 a 1 b 2 c α − 12 a 2 b 1 b 2 c α − 2 a 0 b 1 c ω − 2 a 1 b 2 c ω + 6 a 1 b 1 2 c + 6 a 0 2 b 1 c + 12 a 0 a 1 b 2 c + 12 a 2 b 1 b 2 c + 12 b 1 b 2 γ c 3 + 4 b 1 b 2 c 3 α + 4 b 1 b 2 c 3 = 0 ,

− 12 a 0 b 2 γ c 3 − 6 a 0 b 2 2 c α + 6 a 0 b 2 2 c − 3 b 1 2 γ c 3 + 8 b 2 2 γ c 3 + b 1 2 c 3 α − 8 b 2 2 c 3 α + b 1 2 c 3 − 8 b 2 2 c 3 − 6 b 1 2 b 2 c α − b 2 2 c ω + 6 b 1 2 b 2 c = 0 ,

− 4 a 0 b 1 γ c 3 − 16 a 1 b 2 γ c 3 − 4 a 1 b 2 c 3 α − 4 a 1 b 2 c 3 − 6 a 1 b 2 2 c α − 12 a 0 b 1 b 2 c α + 6 a 1 b 2 2 c + 12 a 0 b 1 b 2 c + 12 b 1 b 2 γ c 3 − 8 b 1 b 2 c 3 α − 8 b 1 b 2 c 3 − 2 b 1 3 c α − 2 b 1 b 2 c ω + 2 b 1 3 c = 0 ,

− 6 a 1 b 1 γ c 3 + 16 a 0 b 2 γ c 3 − 24 a 2 b 2 γ c 3 − 2 a 1 b 1 c 3 α − 8 a 2 b 2 c 3 α − 2 a 1 b 1 c 3 − 8 a 2 b 2 c 3 − 6 a 0 b 1 2 c α − 6 a 2 b 2 2 c α − 6 a 0 2 b 2 c α − 12 a 1 b 1 b 2 c α − 2 a 0 b 2 c ω + 6 a 0 b 1 2 c + 6 a 2 b 2 2 c + 6 a 0 2 b 2 c + 12 a 1 b 1 b 2 c + 2 b 1 2 γ c 3 + 8 b 2 2 γ c 3 − 2 b 1 2 c 3 α + 4 b 2 2 c 3 α − 2 b 1 2 c 3 + 4 b 2 2 c 3 − b 1 2 c ω = 0 ,

− 12 b 1 b 2 γ c 3 + 4 a 1 b 1 b 2 c 3 α + 4 b 1 b 2 c 3 − 6 ω b 1 b 2 2 α + 6 a 0 b 1 b 2 2 c = 0 ,

− 8 b 2 2 γ c 3 + 4 a 0 a 1 γ ω + 4 b 2 2 c 3 α + 4 a 2 b 2 2 c 3 − 2 b 2 3 c α + 2 b 2 3 c = 0 .

Using the well-known Mathematica software to solve the aforementioned system, we find the following families of solutions:

Family 1: For α ≠ 1 and c ≠ 0 :

a 0 = a 0 , a 1 = 0 , a 2 = a 2 , b 1 = 0 , b 2 = − 2 c 2 ( 2 γ − α − 1 ) α − 1 , and ω ≠ 0 .

Substituting the obtained parameters into the general solution (15), we obtain the following family of solutions:

(16) u 1 ( x , t ) = a 0 + a 2 tanh 2 ( c x − ω t ) − 2 c 2 ( 2 γ − α − 1 ) α − 1 coth 2 ( c x − ω t ) .

Family 2: For α ≠ ± 1 and c ≠ 0 :

a 0 = 0 , a 1 = a 1 , a 2 = − 2 c 2 ( 4 γ 2 − α 2 − 2 α − 1 ) 5 ( α 2 − 1 ) , b 1 = 0 , b 2 = − 2 c 2 ( 2 γ − α − 1 ) α − 1 , and ω ≠ 0 .

Therefore, proceeding as in the previous case, the set of solutions for this family is provided by:

(17) u 2 ( x , t ) = a 1 tanh ( c x − ω t ) − 2 c 2 ( 4 γ 2 − α 2 − 2 α − 1 ) 5 ( α 2 − 1 ) tanh 2 ( c x − ω t ) − 2 c 2 ( 2 γ − α − 1 ) α − 1 coth 2 ( c x − ω t ) .

Family 3: For α ≠ 1 and c ≠ 0 :

a 0 = a 0 ≠ 0 , a 1 = a 1 , a 2 = 3 2 a 1 , b 1 = 0 , b 2 = − 2 c 2 ( 2 γ − α − 1 ) α − 1 ,

ω = 8 a 0 γ c 2 α − 8 a 0 γ c 2 + 6 a 0 2 α − 16 γ 2 c 4 + 4 c 4 α 2 + 4 c 4 a 0 ( α − 1 ) .