Integrability characteristics and coherent structures of integrable two-dimensional generalizations of NLS type equations
A recent algorithmic procedure based on truncated Painlev´e expansions is used to derive Lax Pairs, Darboux Transformations, Hirota Tau Functions, and various soliton solutions for integrable (2+1) generalizations of NLS type equations[1]. In particular, diverse classes of solutions are found analogous to the dromion, instanton, lump, and ring soliton solutions derived recently for (2+1) KdV Type Equations, the Nizhnik-Novikov-Veselov Equation, and the Broer-Kaup system. If time permits, we shall also consider an algorithmic method for deriving the integrability characteristics of multicomponent integrable nonlinear PDEs via truncated Painlev´e expansions. A similar method has been well-known for scalar integrable NLPDEs for several years, but no systematic procedure has existed for multicomponent systems. We shall demonstrate that if one uses only one singular manifold, and if one follows the concept of enforcing integrability at each step, then when the system is indeed integrable, one is led immediately and directly to the appropriate Lax Pair, although perhaps in nonlinear form. However, the problem now becomes one of finding a linearization of the nonlinear Lax pair thus found, which so far has had a solution in all cases. The method is illustrated for the NLS, Manakov, and Simple Harmonic Generation systems.