Nonlinear Schrödinger soliton in a time-dependent quadratic potential

We examine the one-soliton solution of the nonlinear Schr\"odinger equation (NLSE) with an external potential of the form of V(x,t)=${\mathit{f}}_{1}$(t)x+${\mathit{f}}_{2}$(t)${\mathit{x}}^{2}$ where ${\mathit{f}}_{1}$(t) and ${\mathit{f}}_{2}$(t) are arbitrary functions of t except that ${\mathit{f}}_{2}$(t) stays above a certain negative value. It is shown that, while the center of the soliton obeys Newton's equation with the potential V(x,t), the internal structure of the soliton is determined by the NLSE of the ``body-fixed'' coordinate system. The soliton structure is found to be independent of ${\mathit{f}}_{1}$(t). The soliton is rigid if ${\mathit{f}}_{2}$(t) is t independent but it can be diffused when ${\mathit{f}}_{2}$(t) varies rapidly. Numerical experiments, however, show that the soliton withstands very rapid variations of ${\mathit{f}}_{2}$(t).

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