Hirota method for the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential

Abstract

In this paper, a Hirota method is developed for applying to the nonlinear Schrö dinger equation with an arbitrary time-dependent linear potential which denotes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensation. The nonlinear Schrö dinger equation is decoupled to two equations carefully. With a reasonable assumption the one-and two-soliton solutions are constructed analytically in the presence of an arbitrary time-dependent linear potential.

Analytical study of the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose–Einstein condensates

“…Moreover, in this paper, the solitonic solutions are derived much more directly and easily than that in Refs. [14,16,18].…”

“…[14,18], we have directly obtained the analytical dark-and bright-solitonic solutions of Eq. (3), which are, respectively, described by expressions (16) and (18). Attention should be paid to these two expressions.…”

“…With Hirota method developed, the one-and two-soliton solutions have been derived effectively in Ref. [18]. Based on the F-expansion method, some Jacobian elliptic function solutions have been presented in Ref.…”

Magnons interaction of spinor Bose-Einstein condensates in an optical lattice

“…Because of the attraction, the magnon cluster tends to be localized, and thus the spin wave becomes unstable. The developing instability causes magnetization localization and brings about a solitary wave [12]. These phenomena are analogous to the ferromagnetism in solid-state physics, but occur with bosons instead of fermions.…”

“…When the potential valley is very deep, the condensates at each lattice site would act as microscopic magnets and interact with each other through the long-range and anisotropic dipole-dipole interaction. These site-to-site dipolar interactions can cause the ferromagnetic phase transition [7,8] leading to a 'macroscopic' magnetization of the condensate array, the spin-wave-like excitation [7][8][9][10] and magnetic soliton [11,12] analogous to the spin-wave and magnetic soliton in a ferromagnetic spin chain. Therefore, the spinor BECs in an optical lattice offers a totally new environment to study spin dynamics in periodic structures.…”