Solitary and traveling wave solutions of the nonlinear Schrodinger equation with quadratic-cubic-quartic nonlinearity

Author : Nikola Petrovic

Abstract :The stabilization of 3-dimensional (3D) solutions to the nonlinear Schrödinger equation (NLSE) with a cubic (third-order, i.e. Kerr) nonlinearity is an important topic of ongoing research in the field of nonlinear optics [1]. The 3D solutions, unlike in the 1D case, are unstable due to critical and supercritical collapse [1]. One of the most common methods to introduce stability in the system is to include competing nonlinearities and the most studied such system has been the NLSE with the cubic-quintic nonlinearity [2]. Recently, a promising approach has been to introduce the Lee-Huang Yang (LHY) correction [3] to the Gross-Pitaevskii equation, a variant of the NLSE describing Bose Einstein condensates which includes an external potential [4]. The LHY correction applied to the NLSE adds a quartic (fourth-order) nonlinear term in the the 3D and the reduced 2D case [5] and a quadratic (second-order) nonlinear term in the 1D case [6]. Systems with quadratic and quartic nonlinearities exist in nonlinear optics [7]. In this paper we will combine the effects to find exact solutions to the NLSE with distributed coefficients and a quadratic-cubic-quartic nonlinearity by generalizing the Jacobi elliptic function (JEF) expansion method developed in [8]. The matching condition for the fourth-order nonlinearity and the second order transverse derivative gives the maximum degree of the JEF to be 2/3 in the ansatz and the principle of harmonic balance gives us solutions which also include the degrees of the JEF of 0 and -2/3. Both solitary and traveling wave solutions are obtained, as well as the so-called chirped solutions, which contain a quadratic dependence in the phase with respect to the transverse variables

Keywords :Nonlinear Schrödinger Equation (NLSE), Cubic Nonlinearity, Supercritical Collapse, Bose-Einstein Condensates, Lee-Huang-Yang (LHY) Correction,

Conference Name :INTERNATIONAL CONFERENCE ON APPLIED PHYSICS AND MATHEMATICS (ICAPM-24)

Conference Place Antalya, Turkey

Conference Date 12th Dec 2024