Soliton solutions to the nonlocal Davey–Stewartson III equation

The soliton solutions on both constant and periodic backgrounds of the nonlocal Davey–Stewartson III equation are derived by using the bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. These solutions are presented as [Formula: see text] Gram-type determinants, with [Formula: see text] a positive integer. Typical dynamics of these soliton solutions are investigated in analytical and graphical aspects. Two types of soliton solutions are generated with different [Formula: see text]. When [Formula: see text] is even, solitons on the constant background can be constructed, whereas solitons appear on the periodic background for odd [Formula: see text]. Under suitable parameter restrictions, we show the regularity of solutions and display all patterns of two- and four-soliton solutions.

Bright-Dark Mixed N-Soliton Solution of Two-Dimensional Multicomponent Maccari System

10.1515/zna-2017-0133 ◽

2017 ◽

Vol 72 (8) ◽

pp. 745-755 ◽

Cited By ~ 2

Author(s):

Zhong Han ◽

Yong Chen

Keyword(s):

Soliton Solution ◽

Reduction Method ◽

Soliton Solutions ◽

Two Dimensional ◽

Kp Hierarchy ◽

Dark Solitons ◽

Long Wave ◽

Special Cases ◽

AbstractBased on the KP hierarchy reduction method, we construct the general bright-dark mixed N-soliton solution of the two-dimensional (2D) (M+1)-component Maccari system comprised of M-component short waves (SWs) and one-component long wave (LW) with all possible combinations of nonlinearities. We firstly consider two types of mixed N-soliton solutions (two-bright-one-dark and one-bright-two-dark solitons in SW components) to the (3+1)-component Maccari system in detail. Then by extending our analysis to the (M+1)-component Maccari system, its general m-bright-(M–m)-dark mixed N-soliton solution is obtained. The formula obtained also contains the general all-bright and all-dark N-soliton solutions as special cases. For the two-bright-one-dark mixed soliton solution of the (3+1)-component Maccari system, it can be shown that solioff excitation and solioff interaction take place in the two SW components supporting bright solitons, whereas the SW component supporting dark solitons and the LW component possess V-type solitary and interaction.

Soliton solutions to a (2+1)-dimensional nonlocal NLS equation

10.21203/rs.3.rs-196261/v1 ◽

2021 ◽

Author(s):

Qiaofeng Huang ◽

Chenzhi Ruan ◽

Jiaxing Huang

Keyword(s):

Boundary Condition ◽

Soliton Solutions ◽

Kp Hierarchy ◽

Nls Equation ◽

Bilinear Method ◽

Back Ground ◽

Nonzero Boundary ◽

Non Local ◽

N Gram

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.

Bright--Dark Solitons in the Space-Shifted Nonlocal Coupled Nonlinear Schrodinger Equation

10.21203/rs.3.rs-1016639/v1 ◽

2021 ◽

Author(s):

Ping Ren ◽

Jiguang Rao

Keyword(s):

Soliton Solutions ◽

Bound State ◽

Dark Soliton ◽

Collision Dynamics ◽

Double Pole ◽

Dark Solitons ◽

Wave Limit ◽

04.20 Jb ◽

Abstract Multiple bright-dark soliton solutions in terms of determinants for the space-shifted nonlocal coupled nonlinear Schro¨dinger (CNLS) equation are constructed by using the bilinear (Kadomtsev-Petviashvili) KP hierarchy reduction method. It is found that the bright-dark two-soliton only occur elastic collisions. Upon their amplitudes, the bright two solitons only admit one pattern whose amplitude are equal, and the dark two solitons have three different non-degenerated patterns and two different degenerated patterns. The bright-dark four-soliton is the superposition of the two-soliton pairs and can generated bound-state solitons. The multiple double-pole bright-dark soliton solutions are generated through the long wave limit of the obtained bright-dark soliton solutions, and their collision dynamics are also investigated.PACS 02.30.Jr · 03.75.Lm · 04.20.Jb · 05.45.Yv

N-soliton and rogue wave solutions of (2+1)-dimensional integrable system with Lax pair

International Journal of Modern Physics B ◽

10.1142/s021797921950317x ◽

2019 ◽

Vol 33 (27) ◽

pp. 1950317 ◽

Cited By ~ 3

Author(s):

Asma Issasfa ◽

Ji Lin

Keyword(s):

Integrable System ◽

Differential Operators ◽

Soliton Solutions ◽

Rational Functions ◽

Rogue Waves ◽

Bilinear Method ◽

The Matrix ◽

Linear Differential ◽

Definition Of

A new generalized (2[Formula: see text]+[Formula: see text]1)-dimensional model, obtained from the Kadomtsev–Petviashvili (KP) equation considered as a system of nonlinear evolution partial differential equations (PDEs), is introduced. With the usage of the Hirota bilinear method and the KP-hierarchy reduction method, N-soliton solutions of the integrable system are constructed. Considering the case of s[Formula: see text]=[Formula: see text]−1 in the linear differential operators, L1 and L2, and a specific set of parameters, two dark solitons, mixed solutions consisting of soliton-type and periodic waves solution are obtained. Based on the particular definition of the matrix elements, one and two rogue waves solutions expressed in terms of rational functions are derived. It is shown that the fundamental rogue waves are line rogue waves, which is different from the property of the moving line solitons of the soliton equations.

General soliton and (semi-)rational solutions of the partial reverse space y-non-local Mel’nikov equation with non-zero boundary conditions

Royal Society Open Science ◽

10.1098/rsos.201910 ◽

2021 ◽

Vol 8 (4) ◽

Author(s):

Heming Fu ◽

Wanshi Lu ◽

Jiawei Guo ◽

Chengfa Wu

Keyword(s):

Boundary Conditions ◽

Reduction Method ◽

The Other ◽

Rational Solutions ◽

Solution Dynamics ◽

Non Local ◽

New Feature ◽

Non Local Equations ◽

General soliton and (semi-)rational solutions to the y-non-local Mel’nikov equation with non-zero boundary conditions are derived by the Kadomtsev–Petviashvili (KP) hierarchy reduction method. The solutions are expressed in N × N Gram-type determinants with an arbitrary positive integer N . A possible new feature of our results compared to previous studies of non-local equations using the KP reduction method is that there are two families of constraints among the parameters appearing in the solutions, which display significant discrepancies. For even N , one of them only generates pairs of solitons or lumps while the other one can give rise to odd numbers of solitons or lumps; the interactions between lumps and solitons are always inelastic for one family whereas the other family may lead to semi-rational solutions with elastic collisions between lumps and solitons. These differences are illustrated by a thorough study of the solution dynamics for N = 1, 2, 3. Besides, regularities of solutions are discussed under proper choices of parameters.

Soliton turbulence in electronegative plasma due to head-on collision of multi solitons

10.1515/zna-2020-0186 ◽

2020 ◽

Vol 75 (12) ◽

pp. 999-1007

Author(s):

Rustam Ali ◽

Anjali Sharma ◽

Prasanta Chatterjee

Keyword(s):

Wave Field ◽

Soliton Solutions ◽

Negative Ions ◽

First Integral ◽

Statistical Properties ◽

Charged Dust ◽

Bilinear Method ◽

Electronegative Plasma ◽

Kdv Equations ◽

First Four Moments

AbstractHead-on interaction of four dust ion acoustic (DIA) solitons and the statistical properties of the wave field due to head-on interaction of solitons moving in opposite direction is studied in the framework of two Korteweg de Vries (KdV) equations. The extended Poincaré–Lighthill–Kuo (PLK) method is applied to obtain two opposite moving KdV equations from an unmagnetized four component plasma model consisting of Maxwellian negative ions, cold mobile positive ions, κ-distributed electrons and positively charged dust grains. Hirota’s bilinear method is adopted to obtain two-soliton solutions of both the KdV equations and accordingly act of soliton turbulence is presented due to head-on collision of four solitons. The amplitude and shape of the resultant wave profile at the point of strongest interaction are obtained. To see the effect of head-on collision on the statistical properties of wave field the first four moments are computed. It is observed that the head-on collision has no effect on the first integral moment while the second, third and fourth moments increase in the dominant interaction region of four solitons, which is a clean indication of soliton turbulence.

Multiple Soliton Solutions for a Variety of Coupled Modified Korteweg–de Vries Equations

Multiple Singular Soliton Solutions

10.5560/zna.2011-0034 ◽

2011 ◽

Vol 66 (10-11) ◽

pp. 625-631

Author(s):

Abdul-Majid Wazwaz

Keyword(s):

Soliton Solutions ◽

Mkdv Equation ◽

Resonance Phenomenon ◽

Hirota’S Bilinear Method ◽

Bilinear Method ◽

Multiple Soliton Solutions ◽

De Vries ◽

Coupled Equation ◽

Singular Soliton ◽

We make use of Hirota’s bilinear method with computer symbolic computation to study a variety of coupled modified Korteweg-de Vries (mKdV) equations. Multiple soliton solutions and multiple singular soliton solutions are obtained for each coupled equation. The resonance phenomenon of each coupled mKdV equation is proved not to exist.

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

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.

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.

Families of exact solutions of the generalized (3+1)-dimensional nonlinear-wave equation

Modern Physics Letters B ◽

10.1142/s0217984918503591 ◽

2018 ◽

Vol 32 (29) ◽

pp. 1850359 ◽

Cited By ~ 8

Author(s):

Wenhao Liu ◽

Yufeng Zhang

Keyword(s):

Wave Equation ◽

Nonlinear Wave ◽

Nonlinear Wave Equation ◽

Soliton Solutions ◽

Rogue Waves ◽

Lump Solution ◽

Rational Solutions ◽

Bilinear Method ◽

The One ◽

Wave Limit

In this paper, the traveling wave method is employed to investigate the one-soliton solutions to two different types of bright solutions for the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear-wave equation, primarily. In the following parts, we derive the breathers and rational solutions by using the Hirota bilinear method and long-wave limit. More specifically, we discuss the lump solution and rogue wave solution, in which their trajectory will be changed by varying the corresponding coefficient or coordinate axis. On the one hand, the breathers express the form of periodic line waves in different planes, on the other hand, rogue waves are localized in time.