Solitons in Plasmas: A Lie Symmetry Approach

Abstract

This paper talks about the stationary solitons for Langmuir waves in plasmas that are described by the Nonlinear Schrödinger's equation with power law nonlinearity. The integration is carried out by the usage of Lie symmetry in presence of perturbation terms.

Modulation of localized solutions for the Schrödinger equation with logarithm nonlinearity

“…Thus, we have δ 1 = 0 and δ 2 = ω 2 sin(ωz)/2. Here the linear coefficient (12) assumes a linear behavior in x, with a periodic modulation in the z-direction while the nonlinear coefficient takes a constant value:…”

“…To be more specific, the LSE appears, for instance, in dissipative systems [3], in nuclear physics [4], in optics [5,6], capillary fluids [7], and even in magma transport [8]. In addition, some important mathematical contributions have been appeared, namely, the existence of stable and localized nonspreading Gaussian shapes [9], dispersion and asymptotic stability features [10], the existence of unique global mild solution [11], the obtention of stationary solutions via Lie symmetry approach [12], and the study of optical solitons with log-law nonlinearity with constant coefficients [13][14][15].…”

Multiple soliton solutions for the integrable couplings of the KdV and the KP equations

“…The ubiquitous KdV equation [1][2][3][4][5][6][7][8][9][10][11][12] in dimensionless variables reads…”

Topological 1-soliton solution of the Biswas–Milovic equation with power law nonlinearity

“…The study of the integrability aspects of nonlinear evolution equations has been going on for the past few decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. There are various techniques that have been developed during this time frame to carry out the integration of these equations.…”

“…There are various techniques that have been developed during this time frame to carry out the integration of these equations. Some of these commonly studied techniques are Inverse Scattering Transform, He's variational principle, Hirota's bilinear method, the Adomian decomposition method, the sub-ODE method, the tanh-coth method, and the Lie symmetry method [7], and there are many more. These methods will integrate nonlinear evolution equations even if they fail the Painleve test of integrability [2,3].…”