Rational solitons of wave resonant-interaction models

摘要

Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be ofnapplicative interest in several areas. Here we consider a system of three coupled wave equations which includesnas special cases the vector nonlinear Schr¨odinger equations and the equations describing the resonant interactionnof three waves. The Darboux-Dressing construction of soliton solutions is applied under the condition that thensolutions have rational, or mixed rational-exponential, dependence on coordinates. Our algebraic constructionnrelies on the use of nilpotent matrices and their Jordan form. We systematically search for all bounded rationaln(mixed rational-exponential) solutions and find a broad family of such solutions of the three wave resonantninteraction equations.