Bifurcation Analysis and Solutions of a Higher-Order Nonlinear Schrödinger Equation

Abstract

The purpose of this paper is to investigate a higher-order nonlinear Schrödinger equation with non-Kerr term by using the bifurcation theory method of dynamical systems and to provide its bounded traveling wave solutions. Applying the theory, we discuss the bifurcation of phase portraits and investigate the relation between the bounded orbit of the traveling wave system and the energy level. Through the research, new traveling wave solutions are given, which include solitary wave solutions, kink wave solutions, and periodic wave solutions.

The First Integral Method for Solving Exact Solutions of Two Higher Order Nonlinear Schrödinger Equations

“…where ξ = pZ − vt. And p, v, λ, w are real parameters, ϕ(ξ) is a real function. Substituting equation (17) into equation 2, and making imaginary and real part zero respectively:…”

“…The theory has been applied to handle various PDEs by many scholars in science and engineering [14][15][16]. Here, we consider the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms [17] and the higher order dispersive Cubic-quintic nonlinear Schrödinger equation [18]:…”

“…For the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, Li et.al employed the bifurcation theory method of dynamical systems to provide the equation bounded traveling wave solutions [17]. Choudhuri and Porsezian analytically solved the high order NLSEs with non-Kerr nonlinearity under some parametric conditions, and he investigated explicitly bright and dark solitary wave solutions and periodic wave solutions [19].…”

Exact Solutions of Two Nonlinear Partial Differential Equations by the First Integral Method

“…DOI: 10.4236/apm.2020.101002 13 Advances in Pure Mathematics [3], bifurcation theory method of dynamic systems [4], Jacobi elliptic function expansion method [5], F expansion method [6], exp-function method [7], Hirota bilinear method [8] and so on.…”

Exact Travelling Wave Solutions of Two Nonlinear Schrödinger Equations by Using Two Methods

“…It can be seen from these fields that the travelling wave solutions of nonlinear evolution equations play an important role in the study. In order to find the exact solutions of nonlinear partial differential Equations (PDEs), pioneers presented the following these methods, such as the first integral method [1], Jacobi elliptic function expansion method [2], F expansion method [3], exp-function method [4], the Kudryashov method [5], the improved ( ) G G ′ -expansion method [6], the tanh-coth method [7], tanh-sech method [8], projective Riccati equation method [9], Kudryashov method [10], sine-cosine method [11], Hirota bilinear method [12], bifurcation theory method of dynamic systems [13] and so on.…”