In this paper, a semi-discrete matrix coupled dispersionless system is presented. A Lax pair is proposed, and the Darboux transformation is employed to construct exact solutions to the semi-discrete ...matrix coupled dispersionless system. These solutions numerically exhibit a variety of exact phenomena, including periodic patterns, breathers, rogue waves, and bright and dark solitons.
•A semi-discrete matrix coupled dispersionless system is presented.•Darboux transformation is employed to construct exact solutions to the system.•Numerical solutions show periodic patterns, breathers, rogue waves, and solitons.
We construct and discuss a semirational, multiparametric vector solution of coupled nonlinear Schrödinger equations (Manakov system). This family of solutions includes known vector Peregrine ...solutions, bright- and dark-rogue solutions, and novel vector unusual freak waves. The vector rogue waves could be of great interest in a variety of complex systems, from optics and fluid dynamics to Bose-Einstein condensates and finance.
Coalgebra symmetry for discrete systems Gubbiotti, G; Latini, D; Tapley, B K
Journal of physics. A, Mathematical and theoretical,
05/2023, Volume:
56, Issue:
20
Journal Article
We introduce a novel family of analytic solutions of the three-wave resonant interaction equations for the purpose of modeling unique events, i.e., "amplitude peaks" which are isolated in space and ...time. The description of these solutions is likely to be a crucial step in the understanding and forecasting of rogue waves in a variety of multicomponent wave dynamics, from oceanography to optics and from plasma physics to acoustics.
We study the Cauchy problem for the focusing nonlinear Schrödinger (fNLS) equation. Using the ∂‾ generalization of the nonlinear steepest descent method we compute the long-time asymptotic expansion ...of the solution ψ(x,t) in any fixed space-time cone C(x1,x2,v1,v2)={(x,t)∈R2:x=x0+vt with x0∈x1,x2,v∈v1,v2} up to an (optimal) residual error of order O(t−3/4). In each cone C the leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton–soliton and soliton–radiation interactions as one moves through the cone. Our results require that the initial data possess one L2(R) moment and (weak) derivative and that it not generate any spectral singularities.
In 1974 Ablowitz, Kaup, Newell, Segur (AKNS) put forward a theoretical framework whereby one can construct evolution equations that are (i) integrable in the sense of existence of infinite number of ...conservation laws and (ii) solvable by the inverse scattering transform. In subsequent years, many physically important integrable evolution equations were identified and the focus of the subject shifted towards methods to find special solutions and enhancing the underlying analysis. The discovery of a new reduction of the original AKNS system and the PT symmetric integrable nonlocal nonlinear Schrödinger (NLS) equation more than forty years later was surprising. Subsequently, additional nonlocal integrable reductions were found allowing nonlocality to be manifested in the time domain as well. This paper reports on yet another novel set of integrable reductions for the original AKNS system and associated new space-time nonlocal NLS type equations with space and time shifts. Integrability and inverse scattering transform are established along with soliton solutions. Their unique properties are discussed along with detailed comparison with the respective standard (non shifted) PT and reverse space-time symmetric NLS equations.
•Nonlocal space-time shifted integrable symmetry reductions.•Space-time shifted nonlocal integrable nonlinear Schrödinger equations.•AKNS scattering problem.•Riemann-Hilbert problems.
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