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.

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).

A two dimensional distributed soliton solution of the Korteweg-de Vries equation

1979 ◽  
Vol 366 (1725) ◽  
pp. 185-204 ◽  
Keyword(s):  
Asymptotic Analysis ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Resonant Interaction ◽  
Two Dimensions ◽  
Discrete Soliton ◽  
Variation Of Parameters ◽  
De Vries ◽  
Interaction Of Solitons ◽  
Discrete Solitons

It is suggested that recent discrete soliton solutions of the Korteweg-de Vries equation in two dimensions can be generalized to solutions which have a continuous variation of parameters. In the particular case of the resonant interaction of solitons, as discussed by Miles, an analytic solution is obtained in which only one parameter is varied. The structure of this solution is examined in detail. Asymptotic solutions for large positive and negative time are construc­ted as well as solutions at large distance for finite time. The asymptotic analysis is found to be similar to that developed by Lighthill for Burgers’ equation. A resonant triad of discrete solitons as described by Miles is shown to develop into a curved soliton. Numerical computations confirm the asymptotic analysis.

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 Physics Informed by Deep Learning: Numerical Solutions of Modified Korteweg-de Vries Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yuexing Bai ◽  
Temuer Chaolu ◽  
Sudao Bilige
Keyword(s):  
Deep Learning ◽  
Prediction Error ◽  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Mkdv Equation ◽  
Dynamic State ◽  
Limited Memory ◽  
De Vries ◽  
Physical Phenomena

In this paper, with the aid of symbolic computation system Python and based on the deep neural network (DNN), automatic differentiation (AD), and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization algorithms, we discussed the modified Korteweg-de Vries (mkdv) equation to obtain numerical solutions. From the predicted solution and the expected solution, the resulting prediction error reaches 1 0 − 6 . The method that we used in this paper had demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations and also opens the way for us to understand more physical phenomena later.

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.

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

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.

Painlevé analysis, auto-Bäcklund transformation and analytic solutions for modified KdV equation with variable coefficients describing dust acoustic solitary structures in magnetized dusty plasmas

2021 ◽  
pp. 2150464
Author(s):  
Shailendra Singh ◽  
S. Saha Ray
Keyword(s):  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Dusty Plasmas ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Painlevé Analysis ◽  
Backlund Transformation ◽  
Painleve Analysis

In this paper, variable coefficients mKdV equation is examined by using Painlevé analysis and auto-Bäcklund transformation method. The proposed equation is an important equation in magnetized dusty plasmas. The Painlevé analysis is used to determine the integrability whereas an auto-Bäcklund transformation technique is being explored to derive unique family of analytical solutions for variable coefficients mKdV equation. New kink–antikink and periodic-kink- type soliton solutions have been determined successfully for the considered equation. This paper shows that auto-Bäcklund transformation method is effective, direct and easy to use, and used to determine the analytic soliton solutions of various nonlinear evolution equations in the field of science and engineering. The results are plotted graphically to signify the potency and applicability of this proposed scheme for solving the above considered equation. The obtained results are in the form of soliton-like solutions, solitary wave solutions, exponential and trigonometric function solutions. Therefore, these solutions help us to understand the potential and physical behaviors of the proposed equation.

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.

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