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 multi-soliton 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. \textbf{110}, 297 (2003)], where only the elementary higher-order zeros were considered. The higher-order solitons corresponding to non-elementary zeros generically describe the simultaneous breakup of a pumping wave $(u_3)$ into the other two components ($u_1$ and $u_2$) and merger of $u_1$ and $u_2$ waves into the pumping $u_3$ wave. The two-soliton solutions corresponding to two simple zeros generically describe the breakup of the pumping $u_3$ wave into the $u_1$ and $u_2$ components, and the reverse process. In the non-generic cases, these two-soliton solutions could describe the elastic interaction of the $u_1$ and $u_2$ waves, thus reproducing previous results obtained by Zakharov and Manakov [Zh. Eksp. Teor. Fiz. \textbf{69}, 1654 (1975)] and Kaup [Stud. Appl. Math. \textbf{55}, 9 (1976)].