REDUCTION PROBLEM FOR FOUR WAVE INTERACTION AND THE CLASSICAL r MATRIX

Abstract

Integrable nonlinear partial differential equations, describing the process of four-wave interaction are derived by means of the reduction procedure. Explicit multisoliton solutions are constructed by formulating a degenerate Reimann–Hilbert problem; while the analytic structure of the Lax operators in the complex plane is used to deduce general Backlund transformations. Conservation laws are derived and the Hamiltonian structure is used to deduce a new form of the classical r matrix, which in turn is used to prove the integrability of our model in case of nontrivial boundary conditions. Lastly the corresponding Quantum R-matrix is derived in the lattice approximation and is seen to be related to an eight-vertex model.