Scattering of quantized solitary waves in the cubic Schrödinger equation

The quantum mechanics for $N$ particles interacting via a $\delta$-function potential in one space dimension and one time dimension is known. The second-quantized description of this system has for its Euler-Lagrange equations of motion the cubic Schrödinger equation. This nonlinear differential equation supports solitary wave solutions. A quantization of these solitons reproduces the weak-coupling limit to the known quantum mechanics. The phase shift for two-body scattering and the energy of the $N$-body bound state is derived in this approximation. The nonlinear Schrödinger equation is contrasted with the sine-Gordon theory in respect to the ideas which the classical solutions play in the description of the quantum states.