General soliton solutions of the NLS
                    +
                    equation under nonvanishing boundary condition

Abstract In this paper, applying the Hirota’s bilinear method and the KP hierarchy reduction method, we obtain the general soliton solutions in the forms of N × N Gram-type determinants to a (2+1)-dimensional non-local nonlinear Schrodinger equation with time reversal under zero and nonzero boundary conditions. The general bright soliton solutions with zero boundary condition are derived via the tau functions of two-component KP hierarchy. Under nonzero boundary condition, we first construct general soliton solutions on periodic back-ground, when N is odd. Furthermore, we discuss typical dynamics of solutions analytically, and graphically.

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Interplay Between Dispersion and Nonlinearity in Femtosecond Soliton Management

Zeitschrift für Naturforschung A ◽

10.1515/zna-2010-6-710 ◽

2010 ◽

Vol 65 (6-7) ◽

pp. 549-554

Author(s):

Ramaswamy Radha ◽

Vaduganathan Ramesh Kumar

Keyword(s):

Pulse Propagation ◽

Group Velocity Dispersion ◽

Soliton Solutions ◽

Higher Order ◽

Order Effects ◽

Nls Equation ◽

Linear Eigenvalue Problem ◽

Third Order Dispersion ◽

Femtosecond Regime ◽

Soliton Management

In this paper, we investigate the inhomogeneous higher-order nonlinear Schr¨odinger (NLS) equation governing the femtosecond optical pulse propagation in inhomogeneous fibers using gauge transformation and generate bright soliton solutions from the associated linear eigenvalue problem. We observe that the amplitude of the bright solitons depends on the group velocity dispersion (GVD) and the self-phase modulation (SPM) while its velocity is dictated by the third-order dispersion (TOD) and GVD. We have shown how the interplay between GVD, SPM, and TOD can be profitably exploited to change soliton width, amplitude (intensity), shape, phase, velocity, and energy for an effective femtosecond soliton management. The highlight of our paper is the identification of ‘optical similaritons’ arising by virtue of higher-order effects in the femtosecond regime.

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A weakly coupled Hirota equation and its rogue waves

Modern Physics Letters A ◽

10.1142/s0217732319501797 ◽

2019 ◽

Vol 34 (22) ◽

pp. 1950179 ◽

Cited By ~ 1

Author(s):

Huijuan Zhou ◽

Chuanzhong Li

Keyword(s):

Optical Fibers ◽

Darboux Transformation ◽

Soliton Solutions ◽

Taylor Series Expansion ◽

Rogue Waves ◽

Nls Equation ◽

Hirota Equation ◽

Weakly Coupled ◽

Parameter Values ◽

Higher Order Dispersion

The Hirota equation, a modified nonlinear Schrödinger (NLS) equation, takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. Its wave propagation is like in the ocean and optical fibers can be viewed as an approximation which is more accurate than the NLS equation. By considering the potential application of two mode nonlinear waves in nonlinear fibers under a certain case, we use the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text] and [Formula: see text] to define a weakly coupled Hirota equation (called Frobenius Hirota equation) including its Lax pair, in this paper. Afterwards, Darboux transformation of the Frobenius Hirota equation is constructed. The Darboux transformation implies the new solutions of ([Formula: see text], [Formula: see text]) generated from the known solution ([Formula: see text], [Formula: see text]). The new solutions ([Formula: see text], [Formula: see text]) provide soliton solutions, breather solutions of the Frobenius Hirota equation. Further, rogue waves of the Frobenius Hirota equation are given explicitly by a Taylor series expansion of the breather solutions. In particular, by choosing different parameter values for the rogue waves, we can get different images.

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On a Riemann–Hilbert problem for the NLS-MB equations

Modern Physics Letters B ◽

10.1142/s0217984921504200 ◽

2021 ◽

pp. 2150420

Author(s):

Leilei Liu ◽

Weiguo Zhang ◽

Jian Xu

Keyword(s):

Hilbert Problem ◽

Soliton Solutions ◽

Coupled System ◽

Soliton Interaction ◽

Nls Equation ◽

Nonzero Boundary ◽

Complex Symmetric ◽

Complex Phenomena ◽

Symmetric Relations ◽

Riemann Hilbert Problem

In this paper, we study a coupled system of the nonlinear Schrödinger (NLS) equation and the Maxwell–Bloch (MB) equation with nonzero boundary conditions by Riemann–Hilbert (RH) method. We obtain the formulae of the simple-pole and the multi-pole solutions via a matrix Riemann–Hilbert problem (RHP). The explicit form of the soliton solutions for the NLS-MB equations is obtained. The soliton interaction is also given. Furthermore, we show that the multi-pole solutions can be viewed as some proper limits of the soliton solutions with simple poles, and the multi-pole solutions constitute a novel analytical viewpoint in nonlinear complex phenomena. The advantage of this way is that it avoids solving the complex symmetric relations and repeatedly solving residue conditions.

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Completeness of the Jost Solutions in the Case of the NLS + Equation with Nonvanishing Boundary Condition

Acta Physica Sinica (Overseas Edition) ◽

10.1088/1004-423x/3/5/001 ◽

1994 ◽

Vol 3 (5) ◽

pp. 321-327

Author(s):

Chen Zong-yun ◽

Huang Nian-ning

Keyword(s):

Boundary Condition ◽

Nls Equation ◽

Jost Solutions

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Perturbations of dark solitons

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2010.0663 ◽

2011 ◽

Vol 467 (2133) ◽

pp. 2597-2621 ◽

Cited By ~ 30

Author(s):

M. J. Ablowitz ◽

S. D. Nixon ◽

T. P. Horikis ◽

D. J. Frantzeskakis

Keyword(s):

Outer Region ◽

Soliton Solutions ◽

Dark Soliton ◽

Nonlinear Damping ◽

Background Intensity ◽

Nls Equation ◽

The Core ◽

Dark Solitons ◽

Direct Perturbation ◽

Inner Region

A direct perturbation method for approximating dark soliton solutions of the nonlinear Schrödinger (NLS) equation under the influence of perturbations is presented. The problem is broken into an inner region, where the core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton that propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated, including linear and nonlinear damping type perturbations.

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The effect of the induced mean flow on solitary waves in deep water

Journal of Fluid Mechanics ◽

10.1017/s0022112097007714 ◽

1998 ◽

Vol 355 ◽

pp. 317-328 ◽

Cited By ~ 11

Author(s):

T. R. AKYLAS ◽

F. DIAS ◽

R. H. J. GRIMSHAW

Keyword(s):

Solitary Wave ◽

Solitary Waves ◽

Deep Water ◽

Water Waves ◽

Wave Equations ◽

Soliton Solutions ◽

Phase Speed ◽

Mean Flow ◽

Nls Equation ◽

Envelope Soliton

Two branches of gravity–capillary solitary water waves are known to bifurcate from a train of infinitesimal periodic waves at the minimum value of the phase speed. In general, these solitary waves feature oscillatory tails with exponentially decaying amplitude and, in the small-amplitude limit, they may be interpreted as envelope-soliton solutions of the nonlinear Schrödinger (NLS) equation such that the envelope travels at the same speed as the carrier oscillations. On water of infinite depth, however, based on the fourth-order envelope equation derived by Hogan (1985), it is shown that the profile of these gravity–capillary solitary waves actually decays algebraically (like 1/x2) at infinity owing to the induced mean flow that is not accounted for in the NLS equation. The algebraic decay of the solitary-wave tails in deep water is confirmed by numerical computations based on the full water-wave equations. Moreover, the same behaviour is found at the tails of solitary-wave solutions of the model equation proposed by Benjamin (1992) for interfacial waves in a two-fluid system.

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