Integrability via Functional Expansion for the KMN Model

Abstract

This paper considers issues such as integrability and how to get specific classes of solutions for nonlinear differential equations. The nonlinear Kundu–Mukherjee–Naskar (KMN) equation is chosen as a model, and its traveling wave solutions are investigated by using a direct solving method. It is a quite recent proposed approach called the functional expansion and it is based on the use of auxiliary equations. The main objectives are to provide arguments that the functional expansion offers more general solutions, and to point out how these solutions depend on the choice of the auxiliary equation. To see that, two different equations are considered, one first order and one second order differential equations. A large variety of KMN solutions are generated, part of them listed for the first time. Comments and remarks on the dependence of these solutions on the solving method and on form of the auxiliary equation, are included.

Solutions of the Bullough–Dodd Model of Scalar Field through Jacobi-Type Equations

“…Therefore, the great advantage brought by the technique used here is related exactly to the freedom of finding BD solutions in correlation with various types of auxiliary equations; this is true, as all of them are reductions of the general Jacobi elliptic equation. The main idea was to find relations among the 12 involved parameters: 4 in the BD Equation (19), 3 in the form of its desired solutions (20), and the 5 parameters {h i , i = 0, 1, 2, 3, 4} from the Jacobi Equation (5). The compatibility condition among the three mentioned equations generated an algebraic system among parameters that can be solved in various situations.…”

“…Substituting (20) along with Equation ( 5) into (19) yields a polynomial equation in φ α α = 0, 6. The fulfillment of this equation imposes that all coefficients vanish for the various powers of φ(ξ).…”

“…Important progress has been achieved and many powerful and effective methods have been proposed for deriving explicit solutions of nonlinear equations: the inverse scattering [8], the Hirota bilinear transformation [9][10][11], the Jacobi elliptic function method [12,13], the generalized Kudryashov method [14,15], the dynamical system approach and the bifurcation method of the phase plane [16], the (G /G)-expansion method [17,18] and its extension to the functional expansion method [19,20], the Lie symmetry method [21][22][23][24][25] and the generalized conditional symmetry approach [26], various extended tanh methods [27][28][29], as well as other tools of investigation for the nonlinear dynamical models [30][31][32][33][34].…”

“…The NPDE solutions will be combinations or series expansions of the auxiliary equation solutions, and they will strongly depend on the choice of this equation. Papers published so far, such as [17][18][19][20] or [35] and the references therein, tend to use a predefined and fixed auxiliary equation and do not investigate how NPDE solutions would depend on the choice of the auxiliary equation. This is actually exactly the main purpose of this study.…”

Bending of solitonic beams modelled by coupled KMN equation in optical birefringent fibers

“…This equation was also re-derived in magnetized plasma system [17] to model nonlinear ion acoustic waves. The detailed exploration of integrable properties and soliton solutions of the equation were also carried out [15,17,18]. The equation was derived in nonlinear optical system by Kundu and Naskar in [19] to explain the arbitrary bending phenomena of optical solitonic beam.…”