Nonlinear Schrödinger soliton in a time-dependent quadratic potential

We examine the one-soliton solution of the nonlinear Schrödinger equation (NLSE) with an external potential of the form of V(x,t)=f1(t)x+f2(t)x2 where f1(t) and f2(t) are arbitrary functions of t except that f2(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 f1(t). The soliton is rigid if f2(t) is t independent but it can be diffused when f2(t) varies rapidly. Numerical experiments, however, show that the soliton withstands very rapid variations of f2(t).