Lie Symmetry Analysis of the Nonlinear Schrödinger Equation with Time Dependent Variable Coefficients

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Preeti Devi ◽  
K. Singh
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Nonlinear Schrodinger Equation ◽  
Symmetry Analysis ◽  
Time Dependent ◽  
Lie Symmetry Analysis ◽  
Nonlinear Schrödinger

scholarly journals Lie symmetry analysis, conservation laws and analytical solutions for chiral nonlinear Schrödinger equation in (2 + 1)-dimensions

2020 ◽  
Vol 25 (3) ◽  
Author(s):  
Jin-Jin Mao ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang ◽  
Xing-Jie Yan
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Lie Symmetry ◽  
Nonlinear Schrodinger Equation ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Lie Symmetry Analysis ◽  
Nonlinear Schrödinger

In this work, we consider the chiral nonlinear Schrödinger equation in (2 + 1)-dimensions, which describes the envelope of amplitude in many physical media. We employ the Lie symmetry analysis method to study the vector field and the optimal system of the equation. The similarity reductions are analyzed by considering the optimal system. Furthermore, we find the power series solution of the equation with convergence analysis. Based on a new conservation law, we construct the conservation laws of the equation by using the resulting symmetries. 

Waves in deep water based on the nonlinear Schrödinger equation with variable coefficients

2014 ◽  
Vol 92 (10) ◽  
pp. 1158-1165
Author(s):  
H.I. Abdel-Gawad
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Variable Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Time Dependent ◽  
Nonlinear Schrödinger ◽  
First Case

It has been shown that progression of waves in deep water is described by the nonlinear Schrödinger equation with time-dependent diffraction and nonlinearity coefficients. Investigation of the solutions is done here in the two cases when the coefficients are proportional or otherwise. In the first case, it is shown that the water waves are traveling at time-dependent speed and are periodic waves, which are coupled to solitons or elliptic waves seen in the noninertial frames. In the inertial frames wave modulation instability is visualized. In the second case, and when the diffraction coefficient dominates the nonlinearity, water waves collapse with unbounded amplitude at finite time. Exact solutions are found here by using the extended unified method together, while presenting a new algorithm for treating nonlinear coupled partial differential equations.

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