arXiv subjects
Publications and source records attributed to Valery Serov.
In a previous article we have proved non-existence of certain "solutions" of the cubically nonlinear Schr\"odinger equation in the general case, and presented solutions in the non-generic case. -- In the present article we describe a further family of solutions enlarging the set of non-generic solutions.
Some years ago, Chen, Pelinovsky, and White claimed existence of certain solutions of the nonlinear Schr\"odinger equation for modelling rogue waves [arXiv: 1909.08165v1 (2019)]. It is the aim of this Comment to outline that this claim is faulty. -- The solutions presented do not satisfy the nonlinear Schr\"odinger equation.
On the basis of analytical results, we present a numerical example that indicates inconsistency of a widely used ansatz with cubically nonlinear Schr\"odinger equation.
For a certain class of solutions of the cubic nonlinear Schödinger equation we prove non-existence in the generic case. In the nongeneric case we present a two-parameter set of solutions, bounded or unbounded, depending on corresponding constraints.
In their interesting article (Physical Review A, Vol. 101, 023843 (2020)) Conforti et al. present doubly periodic (elliptic) solutions of the nonlinear Schrödinger equation, based on an earlier article by Akhmediev et.al. (Theoretical and Mathematical Physics, Vol. 72, 809 (1987)). We present some notes with respect to correctness, completeness, and representation of the solutions obtained.
This work studies the direct and inverse fixed energy scattering problem for two-dimensional Schroedinger equation with rather general nonlinear index of refraction. In particular, using the Born approximation we prove that all singularities of the unknown compactly supported potential from $L^2$-space can be obtained uniquely by the scattering data with fixed positive energy. The proof is based on the new estimates for the Faddeev-Green's function in $L^\infty$-space.
We prove the existence of scattering solutions for multidimensional magnetic Schrödinger equation which belong to the weighted Sobolev space H^1_s (R^n)(n=2,3) with some s < -1/2. As a consequence of this we formulate the direct Born approximation for the magnetic Schrödinger operator. Possible connections with inverse problems (inverse scattering Born approximation) are discussed.