A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation

2003 ◽  
Vol 15 (3) ◽  
pp. 567-574 ◽  
Author(s):  
Engui Fan
Keyword(s):  
Wave Solution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Algebraic Method ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Two Dimensional ◽  
Perturbed Kdv Equation ◽  
Line Soliton

Analytic homoclinic wave and soliton solutions for 2D coupled complex Ginzburg–Landau equations

2018 ◽  
Vol 32 (24) ◽  
pp. 1850286 ◽  
Author(s):  
Qixing Qu ◽  
Li Zhang ◽  
Xiaoyue Liu ◽  
Fenghua Qi ◽  
Xianghua Meng
Keyword(s):  
Wave Solution ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Auxiliary Function ◽  
Function Technique ◽  
Periodic Wave Solution ◽  
Ginzburg Landau ◽  
Kink Wave Solution ◽  
Kink Wave ◽  
Ginzburg Landau Equations

Analytic wave solutions including homoclinic wave, kink wave and soliton solutions for the 2D coupled complex Ginzburg–Landau equations are obtained using the auxiliary function method, Hirota method and the ansatz function technique under certain constraint conditions of coefficients in equations, respectively. The result shows that there exists a kink-wave solution which tends to one and the same periodic wave solution as time tends to infinite.

Constructing Quasi-Periodic Wave Solutions of Differential-Difference Equation by Hirota Bilinear Method

2016 ◽  
Vol 71 (12) ◽  
pp. 1159-1165
Author(s):  
Qi Wang
Keyword(s):  
Difference Equation ◽  
Wave Solution ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Differential Difference Equation ◽  
Hirota Bilinear

AbstractIn the present paper, based on the Riemann theta function, the Hirota bilinear method is extended to directly construct a kind of quasi-periodic wave solution of a new integrable differential-difference equation. The asymptotic property of the quasi-periodic wave solution is analyzed in detail. It will be shown that quasi-periodic wave solution converge to the soliton solutions under certain conditions and small amplitude limit.

Periodic wave solution of the Kundu-Mukherjee-Naskar equation in birefringent fibers via the Hamiltonian-based algorithm

2021 ◽  
Author(s):  
Kang-Jia Wang ◽  
Hong-Wei Zhu
Keyword(s):  
Wave Solution ◽  
Bright Light ◽  
Periodic Wave ◽  
Optical Soliton ◽  
Numerical Results ◽  
Periodic Wave Solution ◽  
Soliton Dynamics ◽  
Good Agreement

Abstract The Kundu-Mukherjee-Naskar equation can be used to address certain optical soliton dynamics in the (2+1) dimensions. In this paper, we aim to find its periodic wave solution by the Hamiltonian-based algorithm. Compared with the existing results, they have a good agreement, which strongly proves the correctness of the proposed method. Finally, the numerical results are presented in the form of 3-D and 2-D plots. The results in this work are expected to shed a bright light on the study of the periodic wave solution in physics.

COEXISTENCE OF MULTIFARIOUS EXACT NONLINEAR WAVE SOLUTIONS FOR GENERALIZED b-EQUATION

2010 ◽  
Vol 20 (10) ◽  
pp. 3193-3208 ◽  
Author(s):  
RUI LIU
Keyword(s):  
Nonlinear Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Solitary Wave Solution ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Wave Solutions ◽  
Special Cases ◽  
Singular Wave ◽  
Nonlinear Wave Solutions

In this paper, we consider the generalized b-equation ut - uxxt + (b + 1)u2ux = buxuxx + uxxx. For a given constant wave speed, we investigate the coexistence of multifarious exact nonlinear wave solutions including smooth solitary wave solution, peakon wave solution, smooth periodic wave solution, single singular wave solution and periodic singular wave solution. Not only is the coexistence shown, but the concrete expressions are given via phase analysis and special integrals. From our work, it can be seen that the types of exact nonlinear wave solutions of the generalized b-equation are more than that of the b-equation. Many previous results are turned to our special cases. Also, some conjectures and questions are presented.

Dispersion relation and wave solution for anharmonic lattices and Korteweg De Vries continua

1973 ◽  
Vol 334 (1596) ◽  
pp. 83-94 ◽  
Keyword(s):  
Dispersion Relation ◽  
Wave Solution ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Perturbation Scheme ◽  
Translational Invariance ◽  
Lennard Jones ◽  
De Vries ◽  
Wave Limit ◽  
Semi Empirical

The set of coupled difference equations describing the dynamics of a monatomic chain is reduced to that of uncoupled anharmonic oscillators through the use of the translational invariance of the equations. Poincaré’s perturbation scheme is then used to obtain a space- and time-wise periodic wave solution together with a nonlinear dispersion relation between frequency, wavenumber and the amplitude. For all wavelengths, the frequency of vibration is observed to increase beyond the corresponding value for the linear case by terms proportional to the square of the amplitude multiplied by the coefficients of the cubic as well as quartic interactions. This solution of the lattice equations suggests a class of solutions of the generalized Korteweg-de Vries (K. de V.) equation which is related to the long wave limit of the nonlinear lattice by means of a semicharacteristic variable stretching transforma­tion. Numerical results for the dispersion relations are presented for the semi-empirical Morse, Born-Meyer and Lennard-Jones potentials. For these physical examples, the con­tributions of the cubic and quartic terms of the interaction potential are found to be of the same order.

scholarly journals The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(m,n)Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Shaoyong Li ◽  
Zhengrong Liu
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We investigate the traveling wave solutions and their bifurcations for the BBM-likeB(m,n)equationsut+αux+β(um)x−γ(un)xxt=0by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-likeB(3,2)equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-likeB(4,2)equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

EXACT SOLITON SOLUTIONS AND QUASI-PERIODIC WAVE SOLUTIONS TO THE FORCED VARIABLE-COEFFICIENT KdV EQUATION

2012 ◽  
Vol 26 (19) ◽  
pp. 1250072 ◽  
Author(s):  
YI ZHANG ◽  
ZHILONG CHENG
Keyword(s):  
Variable Coefficient ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Time Dependent ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Hirota Bilinear ◽  
Exact Soliton Solutions

In this paper, the time-dependent variable-coefficient KdV equation with a forcing term is considered. Based on the Hirota bilinear method, the bilinear form of this equation is obtained, and the multi-soliton solutions are studied. Then the periodic wave solutions are obtained by using Riemann theta function, and it is also shown that classical soliton solutions can be reduced from the periodic wave solutions.

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