Scattering of the vector soliton in coupled nonlinear Schrödinger equation with Gaussian potential

Nonlinear Schrodinger equation (NLSE) is the fundamental equation which describes the wave field envelope dynamics in a nonlinear and dispersive medium. However, if the fields have many components, one should consider the Coupled Nonlinear Schrodinger equation (CNLSE). We considered the interactions of orthogonally polarized and equal-amplitude vector solitons with two polarization directions. In this paper, we focused on the effect of Gaussian potential on the scattering of the vector soliton in CNLSE. The scattering process was investigated by the variational approximation method and direct numerical solution of CNLSE. Analytically, we analyzed the dynamics of the width and center of mass position of a soliton by the variational approximation method. Soliton may be reflected from each other or transmitted through or trapped. Initially, uncoupled solitons may form the coupled state if the kinetic energy of solitons less than the potential of attractive interaction between solitons but when its’ velocity above the critical velocity, the soliton will pass through each other easily. Meanwhile, a direct numerical simulation of CNLSE had been run to check the accuracy of the approximation. The result of the variational model gives a slightly similar pattern with direct numerical simulation of CNLSE by fixing the parameters for both solutions with the same value. The interaction of the vector soliton with Gaussian potential depends on the initial velocity and amplitude of the soliton and the strength of the external potential

Plane wave solution of extended discrete nonlinear Schrödinger equation

In this paper, we considered the extended discrete nonlinear Schrödinger equation (EDNLSE) which includes the nearest neighbour nonlinear interaction in addition to the on-site cubic and quintic nonlinearities. The objective of this study is to investigate the modulational instability of plane matterwave solution in dipolar Bose-Einstein Condensates (BEC) in a periodic optical lattice and to compare the analytical results with numerical. Analytically, the problem is solved by using perturbed solution of the plane wave where the instability of the gain can be obtained. The conditions of the stability of the plane wave had been analysed and confirmed numerically, by applications of Runge- Kutta method. Three specific cases were studied where only cubic-quintic nonlinearity(q = 0) is considered, only quintic-dipolar (alpha = 0) is considered and lastly non-zero for all terms. The numerical results are aligned with the analytical results

Localized solutions of extended discrete nonlinear schrödinger equation

We consider the extended discrete nonlinear SchrÄodinger (EDNLSE) equation which includes the nearest neighbor nonlinear interaction in addition to the on-site cubic and quintic nonlinearities. This equation describes nonlinear excitations in dipolar Bose-Einstein condensate in a periodic optical lattice. We are particularly interested with the existence and stability conditions of localized nonlinear excitations of di®erent types. The problem is tackled numerically, by application of Newton methods and by solving the eigenvalue problem for linearized system near the exact solution. Also the modulational instability of plane wave solution is discussed

Scattering of the vector soliton in coupled nonlinear schrodinger equation with gaussian potential

Nonlinear Schrodinger equation (NLSE) is the fundamental equation which describes the wave field envelope dynamics in a nonlinear and dispersive medium. However, if the fields have many components, one should consider the Coupled Nonlinear Schrodinger equation (CNLSE). We considered the interactions of orthogonally polarized and equal-amplitude vector solitons with two polarization directions. In this paper, we have been focussed on the effect of Gaussian potential on the scattering of the vector soliton in CNLSE. The scattering process was investigated by the variational approximation method and direct numerical solution of CNLSE. Analytically, we analysed the dynamics of the width and center of mass position of a soliton by the variational approximation method. Soliton may be reflected from each other or transmitted through or trapped. Initially, uncoupled solitons may form the coupled state if the kinetic energy of solitons less than the potential of attractive interaction between solitons but when its’ velocity (kinetic energy) above the critical velocity, the soliton will pass through each other easily. Meanwhile, a direct numerical simulation of CNLSE had been run to check the accuracy of the approximation. The result of the variational model will give a slightly similar pattern with direct numerical simulation of CNLSE by fixing the parameters for both solutions with the same value. The interaction of the vector soliton with Gaussian potential depends on the initial velocity and amplitude of the soliton and the strength of the external potential