Nonlinear Schrödinger equation with noncompact isogroup

The properties of the nonlinear Schrödinger equation with noncompact isogroup are investigated. The example of the $U(1,1)$ nonlinear Schrödinger equation reveals the significant differences between this system and the previously considered vector nonlinear Schrödinger equation. The main feature – the large set of admissible boundary conditions on the field functions – leads to a rich spectrum of solutions of the system. Four types of boundary conditions and the corresponding soliton solutions are considered for the $U(1,1)$ model. Quasiclassical quantization of the solitons admits an interpretation in the language of “drops” and “bubbles” as bound states of a large number of bosons of the basic fields subject to the thermodynamic relations for a mixture of gases. The system is a continuous “analog” of the Hubbard model for zero-value boundary conditions, and therefore the paper ends with a discussion of this case.