A mathematical framework is presented to study the evolution of multi-point cumulants in nonlinear dispersive partial differential equations with random input data, based on the theory of weak wave turbulence (WWT). This framework is used to explain how energy is distributed among Fourier modes in the nonlinear Schrödinger equation. This is achieved by considering interactions among four Fourie...

Polynomials in differentiation operators are considered. Joint covariance with respect to Darboux transformations of a pair of such polynomials (Lax pair) as a function of one-field is studied. Methodically, the transforms of the coefficients are equalized to Frechèt differential (first term of the Taylor series on prolonged space) to establish the operator forms. In the commutative (Abelian) c...

Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The global results allow us to study the norm growth of solutions corresponding to the Schrödinger equation term. We use ideas recently introduced to study the class...

In this paper, we prove the nonlinear orbital stability of the stationary traveling wave of the one-dimensional Gross-Pitaevskii equation by using Zakharov-Shabat’s inverse scattering method.