Soliton solutions by means of Hirota bilinear forms

Under investigation in this paper is a variable-coefficient generalized AB system, which is used for modeling the baroclinic instability in the asymptotic reduction of certain classes of geophysical flows. Bilinear forms are obtained, and one-, two- and three-soliton solutions are derived via the Hirota bilinear method. Interaction and propagation of the solitons are discussed graphically. Numerical investigation on the stability of the solitons indicates that the solitons could resist the disturbance of small perturbations and propagate steadily.

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N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0214 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Wen-Xiu Ma

Keyword(s):

Common Factors ◽

Soliton Solutions ◽

Higher Order ◽

Kdv Equations ◽

The Common ◽

Bilinear Formulation ◽

Generalized Kdv Equations ◽

Hirota Bilinear

Abstract We analyze N-soliton solutions and explore the Hirota N-soliton conditions for scalar (1 + 1)-dimensional equations, within the Hirota bilinear formulation. An algorithm to verify the Hirota conditions is proposed by factoring out common factors out of the Hirota function in N wave vectors and comparing degrees of the involved polynomials containing the common factors. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, together with all proofs of the existence of N-soliton solutions to all equations in two classes.

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Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

Zeitschrift für Naturforschung A ◽

10.5560/zna.2012-0044 ◽

2012 ◽

Vol 67 (10-11) ◽

pp. 525-533

Author(s):

Zhi-Qiang Lin ◽

Bo Tian ◽

Ming Wang ◽

Xing Lu

Keyword(s):

Variable Coefficient ◽

Soliton Solutions ◽

Variable Coefficients ◽

Einstein Condensate ◽

Matter Wave ◽

Bilinear Method ◽

Matter Waves ◽

Bose Einstein Condensate ◽

Bose Einstein ◽

Hirota Bilinear

Under investigation in this paper is a variable-coefficient coupled Gross-Pitaevskii (GP) system, which is associated with the studies on atomic matter waves. Through the Painlev´e analysis, we obtain the constraint on the variable coefficients, under which the system is integrable. The bilinear form and multi-soliton solutions are derived with the Hirota bilinear method and symbolic computation. We found that: (i) in the elastic collisions, an external potential can change the propagation of the soliton, and thus the density of the matter wave in the two-species Bose-Einstein condensate (BEC); (ii) in the shape-changing collision, the solitons can exchange energy among different species, leading to the change of soliton amplitudes.We also present the collisions among three solitons of atomic matter waves.

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Lump Solutions, Multi Lump Solutions and More Soliton Solutions of a Novel (2+1)-dimensional Nonlinear Evolution Equation

10.21203/rs.3.rs-996320/v1 ◽

2021 ◽

Author(s):

Hongcai Ma ◽

Shupan Yue ◽

Yidan Gao ◽

Aiping Deng

Keyword(s):

Evolution Equation ◽

Nonlinear Evolution Equation ◽

Nonlinear Evolution ◽

Three Dimensional ◽

Soliton Solutions ◽

Bilinear Method ◽

Test Function ◽

Lump Solutions ◽

Hirota Bilinear ◽

Test Function Method

Abstract Exact solutions of a new (2+1)-dimensional nonlinear evolution equation are studied. Through the Hirota bilinear method, the test function method and the improved tanh-coth and tah-cot method, with the assisstance of symbolic operations, one can obtain the lump solutions, multi lump solutions and more soliton solutions. Finally, by determining different parameters, we draw the three-dimensional plots and density plots at different times.

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Y-shaped soliton solutions for the (2+1)-dimensional bidirectional Sawada–Kotera equation

Modern Physics Letters B ◽

10.1142/s0217984921504881 ◽

2021 ◽

Author(s):

Shuxin Yang ◽

Zhao Zhang ◽

Biao Li

Keyword(s):

Soliton Solutions ◽

Complex Interaction ◽

High Order ◽

Hirota Bilinear Method ◽

Bilinear Method ◽

Dynamic Behaviors ◽

Interaction Solutions ◽

Localized Waves ◽

Hirota Bilinear ◽

Interaction Phenomenon

On the basis of the Hirota bilinear method, resonance Y-shaped soliton and its interaction with other localized waves of (2+1)-dimensional bidirectional Sawada–Kotera equation are derived by introducing the constraint conditions. These types of mixed soliton solutions exhibit complex interaction phenomenon between the resonance Y-shaped solitons and line waves, breather waves, and high-order lump waves. The dynamic behaviors of the interaction solutions are analyzed and illustrated.

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Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves

Modern Physics Letters B ◽

10.1142/s0217984917502682 ◽

2018 ◽

Vol 32 (08) ◽

pp. 1750268 ◽

Cited By ~ 8

Author(s):

Xue-Hui Zhao ◽

Bo Tian ◽

Yong-Jiang Guo ◽

Hui-Min Li

Keyword(s):

Water Waves ◽

Variable Coefficient ◽

Soliton Solutions ◽

Resonance Phenomenon ◽

Bell Polynomials ◽

Bilinear Forms ◽

Linear Superposition ◽

Elastic Interactions ◽

The One ◽

Dimensional Variable

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

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The soliton solution of a modified nonlinear schrödinger equation

Global Journal of Mathematical Analysis ◽

10.14419/gjma.v5i1.7074 ◽

2017 ◽

Vol 5 (1) ◽

pp. 16

Author(s):

Jumei Zhang ◽

Li Yin

Keyword(s):

Schrödinger Equation ◽

Nonlinear Equations ◽

Nonlinear Schrödinger Equation ◽

Soliton Solution ◽

Schrodinger Equation ◽

Soliton Solutions ◽

Nonlinear Schrodinger Equation ◽

Nonlinear Schrödinger ◽

Derivative Method ◽

Hirota Bilinear

Hirota bilinear derivative method can be used to construct the soliton solutions for nonlinear equations. In this paper we construct the soliton solutions of a modified nonlinear Schrödinger equation by bilinear derivative method.

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Bilinear forms and soliton solutions for a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation in an optical fiber

Modern Physics Letters B ◽

10.1142/s0217984920503364 ◽

2020 ◽

Vol 34 (30) ◽

pp. 2050336

Author(s):

Dong Wang ◽

Yi-Tian Gao ◽

Jing-Jing Su ◽

Cui-Cui Ding

Keyword(s):

Schrödinger Equation ◽

Optical Fiber ◽

Nonlinear Schrödinger Equation ◽

Variable Coefficient ◽

Schrodinger Equation ◽

Soliton Solutions ◽

Variable Coefficients ◽

Nonlinear Schrodinger Equation ◽

Bilinear Forms ◽

Dimensional Variable

In this paper, under investigation is a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation, which is introduced to the study of an optical fiber, where [Formula: see text] is the temporal variable, variable coefficients [Formula: see text] and [Formula: see text] are related to the group velocity dispersion, [Formula: see text] and [Formula: see text] represent the Kerr nonlinearity and linear term, respectively. Via the Hirota bilinear method, bilinear forms are obtained, and bright one-, two-, three- and N-soliton solutions as well as dark one- and two-soliton solutions are derived, where [Formula: see text] is a positive integer. Velocities and amplitudes of the bright/dark one solitons are obtained via the characteristic-line equations. With the graphical analysis, we investigate the influence of the variable coefficients on the propagation and interaction of the solitons. It is found that [Formula: see text] can only affect the phase shifts of the solitons, while [Formula: see text], [Formula: see text] and [Formula: see text] determine the amplitudes and velocities of the bright/dark solitons.

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MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

International Journal of Modern Physics B ◽

10.1142/s0217979209053382 ◽

2009 ◽

Vol 23 (25) ◽

pp. 5003-5015 ◽

Cited By ~ 26

Author(s):

XING LÜ ◽

TAO GENG ◽

CHENG ZHANG ◽

HONG-WU ZHU ◽

XIANG-HUA MENG ◽

...

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Symbolic Computation ◽

Bilinear Form ◽

Soliton Solutions ◽

Hirota Bilinear Method ◽

Bilinear Method ◽

Truncated Painlevé Expansion ◽

Hirota Bilinear ◽

Bilinear Equations

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

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Bilinear forms and dark-soliton solutions for a fifth-order variable-coefficient nonlinear Schrödinger equation in an optical fiber

Modern Physics Letters B ◽

10.1142/s0217984916503127 ◽

2016 ◽

Vol 30 (24) ◽

pp. 1650312 ◽

Cited By ~ 14

Author(s):

Chen Zhao ◽

Yi-Tian Gao ◽

Zhong-Zhou Lan ◽

Jin-Wei Yang ◽

Chuan-Qi Su

Keyword(s):

Schrödinger Equation ◽

Optical Fiber ◽

Nonlinear Schrödinger Equation ◽

Variable Coefficient ◽

Schrodinger Equation ◽

Soliton Solutions ◽

Nonlinear Schrodinger Equation ◽

Bilinear Forms ◽

Order Variable ◽

Fifth Order

In this paper, a fifth-order variable-coefficient nonlinear Schrödinger equation is investigated, which describes the propagation of the attosecond pulses in an optical fiber. Via the Hirota’s method and auxiliary functions, bilinear forms and dark one-, two- and three-soliton solutions are obtained. Propagation and interaction of the solitons are discussed graphically: We observe that the solitonic velocities are only related to [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the coefficients of the second-, third-, fourth- and fifth-order terms, respectively, with [Formula: see text] being the scaled distance, while the solitonic amplitudes are related to [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] as well as the wave number. When [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are the constants, or the linear, quadratic and trigonometric functions of [Formula: see text], we obtain the linear, parabolic, cubic and periodic dark solitons, respectively. Interactions between (among) the two (three) solitons are depicted, which can be regarded to be elastic because the solitonic amplitudes remain unchanged except for some phase shifts after each interaction in an optical fiber.

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