Exact soliton solutions of the discrete modified Korteweg–de Vries (mKdV) equation

2010 ◽  
Vol 23 (2) ◽  
pp. 276-284 ◽  
Author(s):  
Yufeng Zhang ◽  
Jianqin Mei ◽  
Y. C. Hon
Keyword(s):  
Soliton Solutions ◽  
Mkdv Equation ◽  
De Vries ◽  
Exact Soliton Solutions

Bright and dark solitons for a generalized Korteweg-de Vries–modified Korteweg-de Vries equation with high-order nonlinear terms and time-dependent coefficients

2011 ◽  
Vol 89 (3) ◽  
pp. 253-259 ◽  
Author(s):  
Houria Triki ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Dark Soliton ◽  
Mkdv Equation ◽  
High Order ◽  
Time Dependent ◽  
Physical Parameters ◽  
Ansatz Method ◽  
Envelope Solitons ◽  
De Vries ◽  
Exact Soliton Solutions

We consider a generalized Korteweg-de Vries–modified Korteweg-de Vries (KdV–mKdV) equation with high-order nonlinear terms and time-dependent coefficients. Bright and dark soliton solutions are obtained by means of the solitary wave ansatz method. The physical parameters in the soliton solutions are obtained as functions of the varying model coefficients. Parametric conditions for the existence of envelope solitons are given. In view of the analysis, we see that the method used is an efficient way to construct exact soliton solutions for such a generalized version of the KdV–mKdV equation with time-dependent coefficients and high-order nonlinear terms.

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

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 ◽  
Multiple Singular Soliton Solutions

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.

scholarly journals Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Jin-Fu Liang ◽  
Xun Wang
Keyword(s):  
Vries Equation ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
Interaction Solutions ◽  
De Vries ◽  
Consistent Riccati Expansion

A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (n) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of n closer to zero (lower bound).

scholarly journals The soliton solutions for semidiscrete complex mKdV equation

2020 ◽  
Vol 34 ◽  
pp. 03002
Author(s):  
Corina N. Babalic
Keyword(s):  
Bilinear Form ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Mkdv Equation ◽  
Complete Integrability ◽  
Nonlinear Schrödinger ◽  
Bilinear Formalism ◽  
De Vries ◽  
Hirota Bilinear

The semidiscrete complex modified Korteweg–de Vries equation (semidiscrete cmKdV), which is the second member of the semidiscrete nonlinear Schrődinger hierarchy (Ablowitz–Ladik hierarchy), is solved using the Hirota bilinear formalism. Starting from the focusing case of semidiscrete form of cmKdV, proposed by Ablowitz and Ladik, we construct the bilinear form and build the multi-soliton solutions. The complete integrability of semidiscrete cmKdV, focusing case, is proven and results are discussed.

scholarly journals Modified Korteweg–de Vries solitons at supercritical densities in two-electron temperature plasmas

2016 ◽  
Vol 82 (2) ◽  
Author(s):  
Frank Verheest ◽  
Carel P. Olivier ◽  
Willy A. Hereman
Keyword(s):  
Soliton Solutions ◽  
Detailed Comparison ◽  
Mkdv Equation ◽  
Perturbation Approach ◽  
Electron Population ◽  
Positive Ions ◽  
Reductive Perturbation ◽  
De Vries ◽  
Temperature Electron ◽  
Two Temperature

The supercritical composition of a plasma model with cold positive ions in the presence of a two-temperature electron population is investigated, initially by a reductive perturbation approach, under the combined requirements that there be neither quadratic nor cubic nonlinearities in the evolution equation. This leads to a unique choice for the set of compositional parameters and a modified Korteweg–de Vries equation (mKdV) with a quartic nonlinear term. The conclusions about its one-soliton solution and integrability will also be valid for more complicated plasma compositions. Only three polynomial conservation laws can be obtained. The mKdV equation with quartic nonlinearity is not completely integrable, thus precluding the existence of multi-soliton solutions. Next, the full Sagdeev pseudopotential method has been applied and this allows for a detailed comparison with the reductive perturbation results. This comparison shows that the mKdV solitons have slightly larger amplitudes and widths than those obtained from the more complete Sagdeev solution and that only slightly superacoustic mKdV solitons have acceptable amplitudes and widths, in the light of the full solutions.

Quasi-Periodic, Periodic Waves, and Soliton Solutions for the Combined KdV-mKdV Equation

2009 ◽  
Vol 64 (9-10) ◽  
pp. 639-645 ◽  
Author(s):  
Emad A.-B. Abdel-Salam
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
De Vries ◽  
Jacobi Elliptic Function Method

By introducing the generalized Jacobi elliptic function, a new improved Jacobi elliptic function method is used to construct the exact travelling wave solutions of the nonlinear partial differential equations in a unified way. With the help of the improved Jacobi elliptic function method and symbolic computation, some new exact solutions of the combined Korteweg-de Vries-modified Korteweg-de Vries (KdV-mKdV) equation are obtained. Based on the derived solution, we investigate the evolution of doubly periodic and solitons in the background waves. Also, their structures are further discussed graphically.

scholarly journals Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Zhe Lin ◽  
Xiao-Yong Wen ◽  
Meng-Li Qin
Keyword(s):  
Nonlinear Optics ◽  
Soliton Solutions ◽  
Elastic Interaction ◽  
Dusty Plasmas ◽  
Mkdv Equation ◽  
Asymptotic State ◽  
Limit States ◽  
Discrete Soliton ◽  
De Vries ◽  
Physical Phenomena

Under investigation is the discrete modified Korteweg-de Vries (mKdV) equation, which is an integrable discretization of the continuous mKdV equation that can describe some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Through constructing the discrete generalized m , N − m -fold Darboux transformation for this discrete system, the various discrete soliton solutions such as the usual soliton, rational soliton, and their mixed soliton solutions are derived. The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically. The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique. Numerical simulations are used to display the dynamical behaviors of some soliton solutions. The results given in this paper might be helpful for better understanding the physical phenomena in plasma and nonlinear optics.

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