Higher‐Order Solitons in the N‐Wave System

Abstract

The soliton dressing matrices for the higher-order zeros of the RiemannHilbert problem for the N-wave system are considered. For the elementary higher-order zero, i.e. whose algebraic multiplicity is arbitrary but the geometric multiplicity is 1, the general soliton dressing matrix is derived. The theory is applied to the study of higher-order soliton solutions in the three-wave interaction model. The simplest higher-order soliton solution is presented. In the generic case, this solution describes the breakup of a higher-order pumping wave into two higher-order elementary waves, and the reverse process. In non-generic cases, this solution could describe (i) the merger of a pumping sech wave and an elementary sech wave into two elementary waves (one sech and the other one higher-order); (ii) the breakup of a higher-order pumping wave into two elementary sech waves and one pumping sech wave; and the reverse processes. This solution could also reproduce fundamental soliton solutions as a special case.

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

“…It is an important fact ͑see Ref. 22, Lemma 2͒ that the sequence of ranks of the projectors P l in the matrix ⌫(k) given by Eq. ͑27͒, i.e., built in the described way, is nonincreasing: rank P n рrank P rϪ1 р¯рrank P 1 ,…”

“…22 for the case of elementary zeros ͑as the equations for the th block resemble analogous equations for a single block corresponding to a pair of elementary zeros͒. For instance, the system ͑63͒ is derived by considering the poles of ⌫(k)⌫ Ϫ1 (k) at kϭ , starting from the highest pole and using the representation ͑59͒-͑60͒ for ⌫ Ϫ1 (k).…”

“…The soliton matrices were previously obtained in two special cases: several pairs of Riemann-Hilbert zeros with equal geometric and algebraic multiplicities, 13 and a single pair of elementary higher-order zeros. 22 In the first case, suppose that the geometric and algebraic multiplicities of n pairs of Riemann-Hilbert zeros ͕(k j ,k j ),1р jрn͖ are ͕r ( j) ,1р jрn͖, respectively. Then the soliton matrices have been given before 13 ͑see also Appendix B in Ref.…”

“…A higher-order zero is called elementary if its geometric multiplicity is 1. 22 This case has been extensively studied in the literature before ͑see Refs. 15, 17, 18, 22͒ for different integrable PDEs.…”

Perturbation theory for bright spinor Bose-Einstein condensate solitons

“…However, for practical calculation of N -soliton effects it is much more convenient to expand a product of rational factors into simple fractions, thereby transforming the product-type expression into a sum-type one [32,40].…”