On integrable wave interactions and Lax pairs on symmetric spaces

Abstract

Multi-component generalizations of derivative nonlinear Schrodinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z_2-graded Lie algebras and A.III symmetric spaces are studied. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed. The latter lead to multi-component integrable equations with CPT-symmetry. Furthermore, the fundamental analytic solutions (FAS) are constructed and the spectral properties of the associated Lax operators are briefly discussed. The Riemann-Hilbert problem (RHP) for the multi-component generalizations of DNLS equation of Kaup-Newell (KN) and Gerdjikov-Ivanov (GI) types is derived. A modification of the dressing method is presented allowing the explicit derivation of the soliton solutions for the multi-component GI equation with both local and nonlocal reductions. It is shown that for specific choices of the reduction these solutions can have regular behavior for all finite x and t. The fundamental properties of the multi-component GI equations are briefly discussed at the end.Comment: 27 pages, LaTe

Nonlocal nonlinear Schrödinger equations and their soliton solutions

“…Existence of this singular behavior of one-soliton solutions of nonlocal NLS equations was also observed in Ref. [10]. Ablowitz and Musslimani have found many other nonlocal integrable equations such as nonlocal modified Korteweg-de Vries equation, nonlocal Davey-Stewartson equation, nonlocal sine-Gordon equation, and nonlocal (2 + 1)-dimensional three-wave interaction equations [2]- [4].…”

Kulish–Sklyanin-type models: Integrability and reductions

“…They are integrable in the sense that they possess Lax representations whose Lax operator is linear in λ with potential Q(x, t) taking value in simple Lie algebra (generalized Zakharov-Shabat system). The solutions for the direct and the inverse scattering problem for such operators by now are well known: see [32,5,1,8,11,15,19,21]. The solutions of these equations can be derived effectively using the dressing Zakharov-Shabat method [35,33,25], see also [8,14,15].…”

Nonlocal Fordy–Kulish equations on symmetric spaces

“…It is highly probable that such a singular structure may arise also in 1-soliton solutions of the system of nonlocal Fordy-Kulish equations. On the other hand it was observed that multi-component derivative NLSE can have regular 1-soliton solutions [10]. In this respect we expect that nonlocal system of nonlocal Fordy equations can have also regular 1-soliton solutions.…”

“…We briefly give the Lax representations of these equations. For more details see [8], [9], [10], [13].…”

“…In this letter we give all possible nonlocal reductions of the Fordy-Kulish system [8] of coupled nonlinear Schrodinger and Fordy system [9] of coupled derivative nonlinear Schrodinger equations on Hermitian symmetric spaces. Nonlocal system equations were also studied recently by Gerdjikov, Grahovski and Ivanov [10]- [12].…”