The inverse-spectral-transform method of solution is shown to be applicable to the physically interesting problem of the nonlinear Schrödinger equation with a general ‘‘potential’’ term, ${\mathrm{iq}}_t$+$q_{\mathrm{xx}}$+2[‖q${\Vert }^2$-F(x)]q=0. The method determines the class of solutions that are symmetric or antisymmetric in x. This is done with the help of a modification of the Ablowitz-Kaup-Newell-Segur and Zakharov-Shabat (AKNS-ZS) formalism incorporating an x- and t-dependent eigenvalue parameter ζ, together with a transformation of variables. In certain physical applications, F(x) describes the inhomogeneity of the medium in which nonlinear wave propagation occurs. The functions F(x) for which the equation is amenable to solution by our method are shown to fall into two classes, depending on whether or not ζ is explicitly t dependent. If it is, we show that F(x) must be a general quadratic function of x. An explicit solution q(x,t) is written down and interpreted for a parabolic potential barrier. If ζ is independent of t, we find that localized solutions with static envelopes can exist for certain other functional forms of F(x). Finally, we comment on the extension of the analysis to explicitly time-dependent potentials or inhomogeneities F(x,t).

• Received 31 May 1984

DOI:https://doi.org/10.1103/PhysRevA.32.1144