General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations

Abstract

We derive the soliton matrices corresponding to an arbitrary number of higher-order normal zeros for the matrix Riemann–Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order solitons. Our soliton matrices explicitly give all higher-order multisoliton solutions to the nonlinear partial differential equations integrable through the matrix Riemann–Hilbert problem. We have applied these general results to the three-wave interaction system, and derived new classes of higher-order soliton and two-soliton solutions, in complement to those from our previous publication [Stud. Appl. Math. 110, 297 (2003)], where only the elementary higher-order zeros were considered. The higher-order solitons corresponding to nonelementary zeros generically describe the simultaneous breakup of a pumping wave (u3) into the other two components (u1 and u2) and merger of u1 and u2 waves into the pumping u3 wave. The two-soliton solutions corresponding to two simple zeros generically describe the breakup of the pumping u3 wave into the u1 and u2 components, and the reverse process. In the nongeneric cases, these two-soliton solutions could describe the elastic interaction of the u1 and u2 waves, thus reproducing previous results obtained by Zakharov and Manakov [Zh. Éksp. Teor. Fiz. 69, 1654 (1975)] and Kaup [Stud. Appl. Math. 55, 9 (1976)].