1-Multisoliton and other invariant solutions of combined KdV-nKdV equation by using Lie Symmetry Approach
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.
Korteweg and Vries derived KdV equation  to model Russell’s phenomenon of solitons like shallow water waves with small but finite amplitudes . 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 , Hirotaâs bilinear method [5, 13], Lie symmetry method[1, 30, 18, 9, 10, 11], Jacobi elliptic function method , Sine-cosine function method , Tanh-coth function method , Weierstrass function method , Homogeneous balance method , Exp-function method , -expansion method , 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
where . We apply Lie symmetry analysis on combined KdV-nKdV equation, first constructed by Wazwaz  using recursion operator . In addition, the combined KdV-nKdV equation (1) possesses the Painlevé property for complete integrability . 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 . 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:
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
where is the group parameter, and the Lie algebra of Eq. (1) is spanned by vector field of the form
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
In the above condition, is termed as -order prolongation  of the infinitesimal generator which is given by
the second summation being over all (unordered) multi-indices , . The coefficient functions of are given by the following expression
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
where and are infinitesimals for the variables and respectively, and is the solution of Eq. (1). Therefore, the associated vector field is
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
Solving the above system of equations, we obtain following infinitesimals for (1) using software Maple,
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
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
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 :
The similarity form suggested by Eq. (13) is given by
with similarity variable
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
where , and are given by Eq. (16).
Case 2: For , Eq. (12) are of the form
The similarity solution to Eq. (21) can be written as
Assume . Without loss of generality, we can assume , then Eq. (34) reduces to
The general solution of Eq. (24) in terms of WeiestrassZeta function as
Case 2A: , Eq. (13) becomes
In this case, we get
Case 3: For , Eq. (13) modified as
The group invariant solution is given as
Hence, from Eq.(34) we found two values of given as
where and are given by Eq. (16).
Case 3A: For , from Eq. (32), we have
Therefore, the similarity transformation method gives
where and is the similarity function. Substituting the value of in Eq. (1) we get reduced ordinary differential equation given as
Again it gives two values of as
The results of the combined KdV-nKdV equation presented in this paper have richer physical structure then earliar outcomes in the literature . 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 .
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 . 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.
The second author sincerely and genuinely thanks Department of Mathematics, SGTB Khalsa College, University of Delhi for financial support.
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