Piecewise smooth localized solutions of Liénard‐type equations with application to NLSE

Abstract

In this work, the rapidly convergent approximation method (RCAM) followed by appropriate modifications is applied to obtain piecewise smooth solutions and conserved quantities of a Liénard‐type equation and some important nonlinear partial differential equations reducible to former one. Explicit parameter dependence of the solution has been sensibly used to determine parameter dependence of the constant of the motion as well as the domain in the parameter space for which the piecewise smooth solution is bounded. Solution of Liénard‐type equation is then used to find piecewise smooth traveling wave solutions and corresponding conserved quantities of nonlinear Schrödinger equation. These findings affirm the efficiency of the RCAM for analytical exercise of variety of nonlinear systems of ordinary/partial differential equations appear as mathematical model of physical processes.

A rapidly convergent approximation scheme for nonlinear autonomous and non-autonomous wave-like equations

“…A few analytical methods such as method based on symmetry analysis (Lie group theoretic approach) [1,2], Hirota bilinear method [3], inverse scattering transformation method [4,5], Prelle-Singer method [6], the method involving Jacobi last multiplier [7], Tanh, Sech, Exp method [8,9], Jacobi elliptic function method [10] and so on, analytical approximation schemes such as homotopy analysis method (HAM) [11,12], Adomian decomposition method (ADM) [13], Fourier transform Adomian decomposition method (FTADM) [14], rapidly convergent approximation method (RCAM) [15][16][17][18][19][20][21][22][23] etc., numerical methods viz. finite difference/element methods [24,25], Galerkin or collocation methods are used to find the solution of mathematical models.…”

New multi-hump exact solitons of a coupled Korteweg-de-Vries system with conformable derivative describing shallow water waves via RCAM

“…These approximate methods are often found to be slowly convergent and unable to provide the close form of the series solution. These problems can be easily tackled by the RCAM [47][48][49][50][51][52][53][54]. In this article, this scheme is used to obtain a new multi-hump travelling wave solution of a coupled Korteweg-de Vries equations with conformable derivative.…”

A rapidly convergent approximation scheme for nonlinear autonomous and non-autonomous wave-like equations

“…A few analytical methods such as method based on symmetry analysis (Lie group theoretic approach) [1,2], Prelle-Singer method [3], the method involving Jacobi last multiplier [4], Tanh, Sech, Exp method [5,6] and so on, analytical approximation schemes such as homotopy analysis method (HAM) [7,8], Adomian decomposition method (ADM) [9], Fourier transform Adomian decomposition method (FTADM) [10], rapidly convergent approximation method (RCAM) [11,12,13,14,15,16,17] etc., numerical methods viz. finite difference/element methods [18,19], Galerkin or collocation methods are used to find the solution of mathematical models.…”