Gaussian Solitary Solution For Several Types Of Logarithmic Nonlinear Schr?ginger Equations

Three types of logarithmic nonlinear Schrodinger equations are investigated in this paper iψ_t+αψ_xx-Vψ+iγlog（ψ/ψ^*）ψ=fψ,iψ_t+αψ_xx-Vψ+βlog|ψ|2ψ=fψ,iψ_t+αψ_xx-Vψ+（iγlog（ψ/ψ^*）+βlog|ψ|²ï¼‰Ïˆ=fψ,which including group velocity dispersion,external potential,logarithmic law nonlinear term and detuning term.We select two potential functions,the standard harmonic oscillator potential V=1/2kx² and the imaginary classicalityenforcing potentialV=iξ[Ψ_xx/Ψ],to investigate the existence of the Gaussian solitary wave solutions for these three kinds of equations.First,for the L-NLSEs under the standard harmonic oscillator potential,when the detuning function f=f（t）,the equations are transformed into several partial differential equations by taking the ansatz Ψ（x,t）=u（x,t）e^{iφ（x,t）},substituting it into the above three equations and separating the real and the imaginary parts respectively.Through the variable separation method and the contradiction method,it is proved that the Gaussian solution does not exist at all in the straight line x=vt（v≠ 0）.Nevertheless,we give the Gaussian solutions in the curve x=v（t）with more universality and application value in a novel way.Then the results are extended into L-NLSEs with（1+n）-dimensions.Secend,for the L-NLSEs under the imaginary classicality-enforcing potential,when the detuning function f=f（ηx-ωt）,the L-NLSEs have Gaussian solutions that propagate in the straight line x=vt（v≠0）,and the expression of Gaussian solutions are given.In addition,these results are also extended to（1+n）-dimensional L-NLSEs.We illustrate the behavior of the Gaussons through numerical simulation and analyze the correlations of Gaussian soliton profile width,wave trajectory,velocity,and range with respect to the relevant parameters,respectively.