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)].