Asymptotic and scattering behaviour for degenerate multi-solitons in the Hirota equation

Abstract

We construct all higher order conserved charges from a general two-dimensional zero curvature condition using a Gardner transformation. Employing two of those charges in the definition of a Hamiltonian allows to view the Hirota equations as an integrable $\mathcal{P}\mathcal{T}$-symmetric extension of the nonlinear Schrödinger equation. We construct new degenerate multi-soliton solutions from Hirota's direct method as well as Darboux-Crum transformations based on Jordan states. We study the properties of these solutions, computing their asymptotic time-dependent displacements and also show that their scattering process has a distinct characteristic behaviour different from the nondegenerate counterparts allowing only for interactions of absorb-emit type.

arXiv comment15 pages, 4 figures

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• Predicting positon solutions of a family of Nonlinear Schrödinger equations through Deep Learning algorithm
• Multicomplex solitons
• Moduli spaces for PT-regularized solitons