2021/04/15 by Wang, Zhaoyu, Chen, Meisen, Fan, Engui
#Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences
paper · doi:10.48550/arxiv.2104.07301
In this paper, we use the ∂ steepest descent method to study the initial value problem for focusing nonlinear Schrödinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we shall characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; further we formulate an appropriate enlarged RH problem; after a series of deformations, the RH problem is transformed into a solvable model. Finally, we obtain the asymptotic expansion of the solution of the fNLS equation in any fixed space-time cone: %as t → ∞, S(x1,x2,v1,v2):=\lbrace (x,t)∈ ℝ2: x=x0+vt, x0∈[x1,x2], v∈[v1,v2]\rbrace. Observing the result indicates that the solution of fNLS equation in this case satisfies the soliton resolution conjecture. The leading order term of this solution includes a high-order pole-soliton whose parameters are affected by soliton-soliton interactions through the cone and soliton-radiation interactions on continuous spectrum. The error term of this result is up to O(t-3/4) which comes from the corresponding ∂ equation.