1-Multisoliton and other invariant solutions of combined KdV-nKdV equation by using Lie Symmetry Approach

# 1-Multisoliton and other invariant solutions of combined KdV-nKdV equation by using Lie Symmetry Approach

Sachin Kumar Department of Mathematics, University of Delhi
Delhi -110007, India
and  Dharmendra Kumar Department of Mathematics, SGTB Khalsa College, University of Delhi-110007, India
###### Abstract.

Lie symmetry method is applied to investigate symmetries of the combined KdV-nKdV equation, that is a new integrable equation by combining the KdV equation and negative order KdV equation. Symmetries which are obtained in this article, are further helpful for reducing the combined KdV-nKdV equation into ordinary differential equation. Moreover, a set of eight invariant solutions for combined KdV-nKdV equation is obtained by using proposed method. Out of the eight solutions so obtained in which two solutions generate progressive wave solutions, five are singular solutions and one multisoliton solutions which is in terms of WeierstrassZeta function.

###### Key words and phrases:
Combined KdV-nKdV equation; Lie symmetry method; Invariant solutions; Similarity reduction.

## 1. Introduction

Korteweg and Vries derived KdV equation [6] to model Russell’s phenomenon of solitons like shallow water waves with small but finite amplitudes [26]. Solitons are localized waves that propagate without change of it’s shape and velocity properties and stable against mutual collision [7, 22]. It has also been used to describe a number of important physical phenomena such as magneto hydrodynamics waves in warm plasma, acoustic waves in an in-harmonic crystal and ion-acoustic waves [17, 2, 8]. A special class of analytical solutions of KdV equation, the so-called traveling waves, for nonlinear evolution equations (NEEs) is of fundamental importance because a lot of mathematical-physical models are often described by such a wave phenomena. Thus, the investigation of traveling wave solutions is becoming more and more attractive in nonlinear science nowadays. However, not all equations posed in these fields are solvable. As a result, many new techniques have been successfully developed by a diverse group of mathematicians and physicists, such as Rational function method [31, 32], Bäcklund transformation method [14], Hirotaâs bilinear method [5, 13], Lie symmetry method[1, 30, 18, 9, 10, 11], Jacobi elliptic function method [12], Sine-cosine function method [27], Tanh-coth function method [15], Weierstrass function method [20], Homogeneous balance method [28], Exp-function method [4], -expansion method [21], etc. But, it is extremely difficult and time consuming to solve nonlinear problems with the well-known traditional methods. This work investigates the combined KdV-nKdV equation

 uxt+6uxuxx+uxxxx+uxxxt+4uxuxt+2uxxut=0, (1)

where . We apply Lie symmetry analysis on combined KdV-nKdV equation, first constructed by Wazwaz [25] using recursion operator [19]. In addition, the combined KdV-nKdV equation (1) possesses the Painlevé property for complete integrability [3]. In this paper, Lie point symmetry generators of the combined KdV-nKdV equation were derived. Similarity reductions and number of explicit invariant solutions for the equation using Lie symmetry method were obtained. All the new invariant solutions of combined KdV-nKdV were analyzed graphically. Also, 1-multisoliton solution obtained in terms of WeierstrassZeta function which appear in classical mechanics such as, motion in cubic and quartic potentials, description of the movement of a spherical pendulum, and in construction of minimal surfaces [29]. Some of the outcomes are interesting in physical sciences and are beautiful in mathematical sciences.

The organization of the paper is as follows. In Sec. 2, we discuss the methodology of Lie symmetry analysis of the general case. In Sec. 3, we obtain infinitesimal generators and the Lie point symmetries of the Eq. (1). In Sec. 4, symmetry reductions and exact group invariant solutions for the combined KdV-nKdV eqation were obtained. In Sec. 5, we discussed all the invariant solutions graphically by Figures 1, 2, 3 and 4. Finally, concluding remarks are summarized in Section 6.

## 2. Method of Lie symmetries

Let us consider a system of partial differential equations as follows:

 Λν(x,u(n))=0,ν=1,2,...,l, (2)

where denotes all the derivatives of of all orders from to . The one-parameter Lie group of infinitesimal transformations for Eq. (2) is given by

 ~xi =xi+ϵξi(x,u)+O(ϵ2);i=1,2,...p, ~uj =uj+ϵϕj(x,u)+O(ϵ2);j=1,2,...q,

where is the group parameter, and the Lie algebra of Eq. (1) is spanned by vector field of the form

 V=p∑i=1ξi(x,u)∂∂xi+q∑j=1ηj(x,u)∂∂uj (3)

A symmetry of a partial differential equation is a transformation which keeps the solution invariant in the transformed space. The system of nonlinear PDEs leads to the following invariance condition under the infinitesimal transformations

 Pr(n)V[Λν(x,u(n))]=0,ν=1,2,...l% along with Λ(x,u(n))=0

In the above condition, is termed as -order prolongation [16] of the infinitesimal generator which is given by

 Pr(n)V=V+q∑j=1∑JηJj(x,u(n))∂∂ujJ (4)

the second summation being over all (unordered) multi-indices , . The coefficient functions of are given by the following expression

 ηJj(x,u(n))=DJ(ηj−p∑i=1ξiuji)+p∑i=1ξiujJ,i (5)

where , and denotes total derivative.

## 3. Lie symmetry analysis for the combined KdV-nKdV equation

In this section, authors explained briefly all the steps of the STM method to keep this work self-confined. The Lie symmetries for the Eq. (1) have generated and then its similarity solutions are found. Therefore, one can consider the following one-parameter () Lie group of infinitesimal transformations

 ~x =x+ϵξ(x,t,u)+O(ϵ2), ~t =t+ϵτ(x,t,u)+O(ϵ2), ~u =u+ϵη(x,t,u)+O(ϵ2), (6)

where and are infinitesimals for the variables and respectively, and is the solution of Eq. (1). Therefore, the associated vector field is

 V=ξ(x,t,u)∂∂x+τ(x,t,u)∂∂t+η(x,t,u)∂∂u. (7)

Lie symmetry of Eq. (1) will be generated by Eq. (7). Use fourth prolongation gives rise to the symmetry condition for Eq. (1) as follows:

 ηxt+6ηxuxx+6ηxxux+ ηxxxx+ηxxxt+4ηxuxt + 4ηxtux+2ηtuxx+2ηxxut=0, (8)

where , and are the coefficient of , values are given in many references [16, 1]. Incorporating all the expressions into Eq. (3), and then equating the various differential coefficients of to zero, we derive following system of Eight determining equation

 ξu=ξxx=0,ξx+ξt=τt,τx=τu=0, ξx+2ηx=ξx+ηu=0, ξx+12τt=ηt. (9)

Solving the above system of equations, we obtain following infinitesimals for (1) using software Maple,

 ξ=(x−t)a1+a2+f(t),τ=f(t),η=(t−x2−u)a1+a3+12f(t), (10)

where and are arbitrary constants whereas is an arbitrary function.

The symmetries under which Eq. (1) is invariant can be spanned by the following four infinitesimal generators if we assume , a constant Then all of the infinitesimal generators of Eq. (1) can be expressed as

 V=a1V1+a2V2+a3V3+cV4.

where

 V1 = 2(x−t)∂∂x+(t−x2−u)∂∂u, V2 = ∂∂x, V3 = ∂∂u, V4 = ∂∂x+∂∂t+∂∂u. (11)

The vector field yield commutation relations through the Table 1. The th entry in Table 1 is the Lie bracket . Table 1 is skew - symmetric with zero diagonal elements. Table 1 shows that the generators and are linearly independent. Thus, to obtain the similarity solutions of Eq. (1), the corresponding associated Lagrange system is

 dxξ(x,t,u)=dtτ(x,t,u)=duη(x,t,u). (12)

## 4. Invariant solutions of the combined KdV-nKdV equation

To proceed further, selection of and by assigning the particular values to ’s , provide new physically meaningful solutions of Eq. (1). In order to obtain symmetry reductions and invariant solutions, one has to solve the associated Lagrange equations given by (12). Now, let us discuss the following particular cases for various forms of :

Case 1: For , then Eqs. (10) and (12) gives

 dx(x−t)a1+a2+f(t)=dtf(t)=du(t−x2−u)a1+a3+12f(t). (13)

The similarity form suggested by Eq. (13) is given by

 u=α+14(3t−x)+e−2a1βtan−1(βT)U(X), (14)

with similarity variable

 X=(x−t+A)e−2a1βtan−1(βT), (15)

where

 α=a2+4a34a1,β=1√4ac−b2,A=a2a1,T=2at+b. (16)

Inserting the value of from Eq. (14) into Eq. (1), we get the following fourth order ordinary differential equation

 XUXXXX+4UXXX+6XUXUXX+2UUXX+8U2X=0, (17)

where is given by Eq. (15) and , etc.

Any how, we could not find the general solution of Eq. (17) still two particular solutions are found as below

 U(X)=c1andU(X)=c2X, (18)

where and are arbitrary constants. Thus, from Eqs. (14) and (18), we get two invariant solutions of Eq. (1) given below

 u(x,t) =α+14(3t−x)+c1e−2a1βtan−1(βT), (19) u(x,t) =α+14(3t−x)+c2A+x−t, (20)

where , and are given by Eq. (16).

Case 2: For , Eq. (12) are of the form

 dxc+a2=dtc=duc2+a3. (21)

The similarity solution to Eq. (21) can be written as

 u=(12+a3c)t+U(X), (22)

where is a similarity variable.
Inserting the value of from Eq. (22) in Eq. (1), we get the fourth order ordinary differential equation in

 a2UXXXX+(a2−2a3+6a2UX)UXX=0. (23)

Assume . Without loss of generality, we can assume , then Eq. (34) reduces to

 UXXXX+6UXUXX=0. (24)

The general solution of Eq. (24) in terms of WeiestrassZeta function as

 U(X)=c3+(−1)13223WeierstrassZeta[α1(x,t),α2], (25)

where and and are arbitrary constants. From Eq. (25) with Eq. (22), we have another invariant solution of Eq. (1)

 u(x,t)=c3+(12+a3c)t+(−1)13223WeierstrassZeta[α1(x,t),α2]. (26)

Case 2A: , Eq. (13) becomes

 dx(x−t)a1+c=dtc=du(t−x2−u)a1+a3+c2. (27)

In this case, we get

 u=a3a1+14(3t−x)+e−a1ctU(X), (28)

where . Substituting the value of in Eq. (1), again we obtain the same fourth order ordinary differential equation (17). Some particular solutions are given below

 U(X)=c6,U(X)=c7X, (29)

where and are arbitrary constants. Therefore, using Eq. (29) in Eq. (28), we obtain the following two exact solutions for Eq. (1)

 u(x,t) =a3a1+14(3t−x)+c6e−a1ct, (30) u(x,t) =a3a1+14(3t−x)+c7x−t. (31)

Case 3: For , Eq. (13) modified as

 dx(x−t)a1+a2=dt0=du(t−x2−u)a1+a3. (32)

The group invariant solution is given as

 u =a1x(x−4t)−4a3x+4U(T)4a1(t−x)−4a2, (33)

with . Substituting the value of from Eq. (33) into Eq. (1), we get the following reduced ordinary differential equation

 [3a21(t2−1)−2a1(a2t−2a3t+2U(t))−a2(a2+4a3)][3a1t−a2+2a3−2U′(t)]=0. (34)

Hence, from Eq.(34) we found two values of given as

 U(T) =3a21(t2−1)−2(a2−2a3)a1t−a2(a2+4a3)4a1, (35) U(T) =14(3a1t2−2a2t+4a3t+4c8), (36)

where is an arbitrary constant of integration. Here, Eq. (35) gives same solution as Eq. (20) with . Also, using Eq. (36) in Eq. (33) gives new invariant solution of Eq. (1) as

 u(x,t)=4(a3(t−x)+c8)+a1(3t2−4tx+x2)−2a2t4a1(t−x)−4a2, (37)

where and are given by Eq. (16).

Case 3A: For , from Eq. (32), we have

 dx(x−t)a1=dt0=du(t−x2−u)a1+a3. (38)

Therefore, the similarity transformation method gives

 u=4xa1−x(x−4t)a3−4U(T)4(a1+(x−t)a3), (39)

where and is the similarity function. Substituting the value of in Eq. (1) we get reduced ordinary differential equation given as

 [a3(3a3(t2−1)−4U(t))+2a3a1t−5a21][3a3t+a1−2U′(t)]=0. (40)

Again it gives two values of as

 U(T) =3a23(t2−1)+2a3a1t−5a214a3, (41) U(T) =14(2a1t+3a3t2+4c9), (42)

where is constant of integration. Here, Eq. (41) gives same solution as Eq. (20). Also, using (42) with (39) gives another new invariant solution of (1) as

 u(x,t) =2a1(2x−t)−a3(3t2−4tx+x2)−4c94(a3(x−t)+a1). (43)

Case 4: For , from Eq. (32) we get similarity form suggested by Eq.(13) is given by

 u=α+14(3t−x)+ea12t2U(X), (44)

with similarity variable and is given by (16). In this case, we get same ordinary differential equation as Eq. (17), one particular solution is as follows

 U(X)=c10, (45)

where is arbitrary constant. Thus, from Eqs. (44) and (45) we get another new invariant solution of Eq. (1) given below

 u(x,t) =α+14(3t−x)+c10ea12t2. (46)

## 5. Discussion

The results of the combined KdV-nKdV equation presented in this paper have richer physical structure then earliar outcomes in the literature [25]. The recorded results are significant in the context of nonlinear dynamics, physical science, mathematical physics etc. The invariant solutions obtained can illustrate various dynamic behaviour due to existence of arbitrary constants. The nonlinear behaviour of the results are analyzed in the following manner:
Figure 1: Fig 1(a) shows progressive wave soluion in Eq. (19) for . Fig 1(b) shows presence of singularity in the plane in Eq. (20) with values .
Figure 2: The evolution profiles of 1-Multisoliton solution is given by Eq. (26) as shown in this Figure. We have recorded the physical nature with variation in parameters. Fig 2(a) For and a plot of WeiestrassZeta function is shown; Fig 2(b) For and only few solitons are shown; Fig 2(c) Front orthographic projection is shown for and ; Fig 2(d) Front orthographic projection is shown for and .
Figure 3: Fig 3(a) For and , Eq. (30) exhibits progressive wave; Fig 3(b), shows singularity in planes for Eqs. (31) for and ; Fig 3(c, d) shows singularity wave profile for Eq. (37) and Eq. (43) which explain the transition of nonlinear behaviour in the form of opposite rotatory folded sheets.
Figure 4: exhibits singularity near but asymptotic structures is observed near for parameters .

## 6. Conclusion

In this paper, the similarity reductions and invariant solutions for the combined KdV-nKdV are presented. This paper obtained the 1-multisoliton and other invariant solution of the equation. The method that was used to obtain the exact group invariant solutions is the Lie symmetry analysis approach. All the solutions are different from earlier work which have been obtained by Wazwaz [25]. Eventually, the stucture of combined KdV-nKdV equation is an non-trivial one, which can be clearly seen from the graphically results of invariant solutions. The Lie symmetry analysis method extracts the new forms of analytic solutions which are of physical importance such as condensed matter physics and plasma physics.

## Acknowledgment

The second author sincerely and genuinely thanks Department of Mathematics, SGTB Khalsa College, University of Delhi for financial support.

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