Time‐dependent Schrödinger equations: Symmetry breaking, separation of variables, and nonlinear effects

Nonlinear equations are important for the description of dissipative processes and soliton‐related phenomena. One of the principal aims of the theory of symmetry breaking in quantum mechanics is to provide a systematic method for introducing interactions which reduce the symmetry of a system. Such a method is applied here to compute the general form of the symmetry‐breaking interaction F ( x , t , Ψ, Ψ*) for the time‐dependent Schrödinger equation in one spatial dimension\documentclass{article}\pagestyle{empty}\begin{document}$$ \left[{\partial _{xx} + 2i\partial_t - 2g_2(t)x^2 - 2g_1(t)x - 2g_0 (t)} \right]\psi (x,t) = F(x,t,\psi,\psi *). $$\end{document} For F = 0, it has been shown that the Lie algebra of space–time invariances of this equation is S 1 = s 1 (2,IR)□ w 1 , the Schrödinger algebra. Following the method of Boyer, Sharp, and Winternitz, all conjugacy classes of subalgebras of S 1 are given. For each subalgebra, the most general form of the interaction term F ( x , t , Ψ, ψ*) is constructed. The potential F then reduces the symmetry from S 1 to the considered subalgebra. Furthermore, the one‐dimensional subalgebras of S 1 obtained above partition S 1 into orbits of operators. To each orbit there corresponds a coordinate system in which the above equation separates variables. A partial resolution of the solutions of the above equation has been obtained by exploiting this relation. The algebraic approach to symmetry breaking yields a rich variety of interaction terms F , which appear to generalize the nonlinear Schrödinger–Langevin–Kostin equation for nonconservative systems and the so‐called nonlinear Schrödinger equation. A criterion in terms of the existence or nonexistence of Bäcklund transformations is conjectured in order to distinguish between dissipative and soliton equations.