Soliton resonances in a generalized nonlinear Schrödinger equation

Abstract

It is shown that a generalized nonlinear Schrödinger equation proposed by Malomed and Stenflo admits, for a specific range of parameters, resonant soliton interaction. The equation is transformed to the 'resonant' nonlinear Schrödinger equation, as originally introduced to describe black holes in a Madelung fluid and recently derived in the context of uniaxial wave propagation in a cold collisionless plasma. A Hirota bilinear representation is obtained and soliton solutions are thereby derived. The one-soliton solution interpretation in terms of a black hole in two-dimensional spacetime is given. For the twosoliton solution, resonant interactions of several kinds are found. The addition of a quantum potential term is considered and the reduction is obtained to the resonant NLS equation.

Envelope soliton resonances and broer-kaup-type non-madelung fluids

“…The NLS equation with such a term was derived for the Jackiw-Teitelboim gravity in [6], [15], [16] and in plasma physics in [17]. The last two terms were introduced for a Hamiltonian generalization of the NLS equation by Malomed and Stenflo [18] and were studied for the resonance behavior in [19]. If we diagonalize dispersion term (8), then we obtain the standard linear dispersion…”

Chiral resonant solitons in Chern–Simons theory and Broer–Kaup type new hydrodynamic systems

“…As an integrable capillarity model in [12]. It was studied recently as an integrable deformation of dispersion for generic envelope equation of nonlinear Schrodinger type [7].…”

“…The capillarity model [12], and information theory with Fisher measure of maximal uncertainty, [8]. It was studied recently as integrable deformation of dispersion for generic envelope equation of nonlinear Schrodinger type [7]. Subsequently, the influence of this potential on anyons in 2 + 1 dimensions has been studied [4], and the Abelian Chern-Simons gauge field interacting with NLS has been represented as a planar Madelung fluid [6], where the Chern-Simons Gauss law has the simple physical meaning of creation of the local vorticity for the flow.…”

“…Hence the chiral solitons appear as solutions of the derivative NLS with quantum potential [5]. The last one by the Madelung transform is represented as the derivative Reaction-Diffusion (DRD) system, arising in SL (2, R) Kaup-Newell hierarchy, and giving rise to the resonant soliton phenomena [7].…”

Резонансы Солитонов Огибающих И Немаделунговы Жидкости Типа Броера - Каупа

“…НУШ с таким членом было выведено для гравитации Джакива-Тейтельбаума в ра-ботах [6], [15], [16] и в физике плазмы в работе [17]. Последние два члена были введены для гамильтонова обобщения НУШ, предложенного Маломедом и Стен-фло [18], а их резонансное поведение изучалось в работе [19]. Если мы диагонализуем дисперсионный член (8), то для эллиптического случая g = EG − F 2 > 0 получим обычную линейную дисперсию…”