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

2019 ◽  
Vol 33 (27) ◽  
pp. 1950317 ◽  
Author(s):  
Asma Issasfa ◽  
Ji Lin
Keyword(s):  
Integrable System ◽  
Differential Operators ◽  
Soliton Solutions ◽  
Rational Functions ◽  
Rogue Waves ◽  
Bilinear Method ◽  
Kp Hierarchy Reduction 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.

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

2018 ◽  
Vol 32 (29) ◽  
pp. 1850359 ◽  
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.

Localized waves and interaction solutions to an extended (3+1)-dimensional Kadomtsev–Petviashvili equation

2020 ◽  
Vol 34 (06) ◽  
pp. 2050076 ◽  
Author(s):  
Han-Dong Guo ◽  
Tie-Cheng Xia ◽  
Wen-Xiu Ma
Keyword(s):  
Three Dimensional ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Dark Soliton ◽  
Complex Conjugate ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Parameter Constraints ◽  
Localized Waves

In this paper, an extended (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied via the Hirota bilinear derivative method. Soliton, breather, lump and rogue waves, which are four types of localized waves, are obtained. N-soliton solution is derived by employing bilinear method. Then, line or general breathers, two-order line or general breathers, interaction solutions between soliton and line or general breathers are constructed by complex conjugate approach. These breathers own different dynamic behaviors in different planes. Taking the long wave limit method on the multi-soliton solutions under special parameter constraints, lumps, two- and three-lump and interaction solutions between dark soliton and dark lump are constructed, respectively. Finally, dark rogue waves, dark two-order rogue waves and related interaction solutions between dark soliton and dark rogue waves or dark lump are also demonstrated. Moreover, dynamical characteristics of these localized waves and interaction solutions are further vividly demonstrated through lots of three-dimensional graphs.

A COMPLETELY INTEGRABLE SYSTEM OF COUPLED MODIFIED KdV EQUATIONS

2010 ◽  
Vol 19 (01) ◽  
pp. 145-151 ◽  
Author(s):  
ABDUL-MAJID WAZWAZ
Keyword(s):  
Integrable System ◽  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Kdv Equations ◽  
Multiple Soliton Solutions ◽  
Modified Kdv Equations ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

In this work, we study a system of coupled modified KdV (mKdV) equations. Multiple soliton solutions and multiple singular soliton solutions are derived by using the Hirota's bilinear method and the Hietarinta approach. The resonance phenomenon is examined.

Homoclinic breather waves, rouge waves and multi-soliton waves for a (2+1)-dimensional Mel’nikov equation

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Na Liu
Keyword(s):  
Design Methodology ◽  
Evolution Equations ◽  
Dynamic Properties ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Short Wave ◽  
Nonlinear Evolution Equations ◽  
Bilinear Method ◽  
Content Type ◽  
Extreme Behavior

Purpose The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets. Design/methodology/approach The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method. Findings The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures. Research limitations/implications This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations. Originality/value The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

scholarly journals Novel Localized Waves and their Interaction Solutions for a Dimensionally Reduced (2+1)-dimensional Boussinesq Equation from N-soliton Solutions

2021 ◽  
Author(s):  
Dipankar Kumar ◽  
Md. Nuruzzaman ◽  
Gour Chandra Paul ◽  
Ashabul Hoque
Keyword(s):  
Water Waves ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Rogue Waves ◽  
Ocean Engineering ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Wave Limit ◽  
Localized Waves

Abstract The Boussinesq equation (BqE) has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave propagation, wave over-topping, inundation, and near-shore wave process in which nonlinearity and dispersion effects are taken into consideration. The study deals with the dynamics of localized waves and their interaction solutions to a dimensionally reduced (2 + 1)-dimensional BqE from N-soliton solutions with the use of Hirota’s bilinear method (HBM). Taking the long-wave limit approach in coordination with some constraint parameters in the N-soliton solutions, the localized waves (i.e., soliton, breather, lump, and rogue waves) and their interaction solutions are constructed. The interaction solutions can be obtained among localized waves, such as (i) one breather or one lump from the two solitons, (ii) one stripe and one breather, and one stripe and one lump from the three solitons, and (iii) two stripes and one breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two lumps from the four solitons. It is to be found that all interactions among the solitons are elastic. The energy, phase shift, shape, and propagation direction of these localized waves and their interaction solutions can be influenced and controlled by the involved constraint parameters. The dynamical characteristics of these localized waves and their interaction solutions are demonstrated through some 3D and density graphs. The outcomes achieved in this study can be used to illustrate the wave interaction phenomena in shallow water.

scholarly journals Linear Differential Operators Invertible in the Integro-differential Sense

2019 ◽  
pp. 20-33
Author(s):  
V. N. Chetverikov
Keyword(s):  
Differential Operator ◽  
Integral Operator ◽  
Differential Operators ◽  
Spatial Variable ◽  
Diagonal Form ◽  
Operator Equations ◽  
Evolutionary Systems ◽  
Linear Differential Operators ◽  
The Matrix ◽  
Linear Differential

The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.

Soliton solutions to the nonlocal Davey–Stewartson III equation

2020 ◽  
pp. 2150026
Author(s):  
Heming Fu ◽  
Chenzhen Ruan ◽  
Weiying Hu
Keyword(s):  
Positive Integer ◽  
Reduction Method ◽  
Soliton Solutions ◽  
Regularity Of Solutions ◽  
Kp Hierarchy ◽  
Bilinear Method ◽  
Suitable Parameter ◽  
Kp Hierarchy Reduction Method ◽  
Kp Hierarchy Reduction

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.

Linear quasi-differential operators in locally integrable spaces on the real line

2000 ◽  
Vol 130 (4) ◽  
pp. 671-698 ◽  
Author(s):  
R. R. Ashurov ◽  
W. N. Everitt
Keyword(s):  
Banach Spaces ◽  
Real Line ◽  
Differential Operators ◽  
Linear Topological Space ◽  
Topological Spaces ◽  
The Real ◽  
Conjugate Space ◽  
Linear Differential ◽  
Definition Of ◽  
Locally Convex

The theory of ordinary linear quasi-differential expressions and operators has been extensively developed in integrable-square Hilbert spaces. There is also an extensive theory of ordinary linear differential expressions and operators in integrable-p Banach spaces.However, the basic definition of linear quasi-differential expressions involves Lebesgue locally integrable spaces on intervals of the real line. Such spaces are not Banach spaces but can be considered as complete locally convex linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete locally convex linear topological space, but now with the topology derived as a strict inductive limit.This paper develops the properties of linear quasi-differential operators in a locally integrable space and the first conjugate space. Conjugate and preconjugate operators are defined in, respectively, dense and total domains.

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

2021 ◽  
Author(s):  
Qiaofeng Huang ◽  
Chenzhi Ruan ◽  
Jiaxing Huang
Keyword(s):  
Boundary Condition ◽  
Soliton Solutions ◽  
Kp Hierarchy ◽  
Nls Equation ◽  
Bilinear Method ◽  
Kp Hierarchy Reduction 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.

Rational and semi-rational solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

2019 ◽  
Vol 33 (25) ◽  
pp. 1950296 ◽  
Author(s):  
Ya-Si Deng ◽  
Bo Tian ◽  
Yan Sun ◽  
Chen-Rong Zhang ◽  
Cong-Cong Hu
Keyword(s):  
Water Waves ◽  
Boussinesq Equation ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Different Shapes ◽  
Lump Wave ◽  
Kink Soliton

Nonlinear waves are seen in nature, such as the water waves and plasma waves. Investigated in this paper is a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Based on the bilinear method, we get the rational solutions, which are different from the published ones, semi-rational solutions and breather-type kink soliton solutions. Through the rational solutions, we observe two types of waves: the lump waves and line rogue waves. The semi-rational solutions depict two types of interactions: (1) The fusion or fission between the lump wave and soliton; (2) The interaction between the line rogue wave and soliton. During the interaction between the line rogue wave and soliton, the line rogue wave evolves with three different shapes: the bright rogue waves, bright–dark rogue waves and dark rogue waves. Via the breather-type kink soliton solutions, we observe the breather-soliton mixture.

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