Two new integrable modified KdV equations, of third-and fifth-order, with variable coefficients: multiple real and multiple complex soliton solutions

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Variable Coefficients ◽  
Kdv Equations ◽  
Modified Kdv Equations ◽  
Fifth Order ◽  
Multiple Complex Soliton Solutions

Two integrable third-order and fifth-order KdV equations with time-dependent coefficients

2019 ◽  
Vol 29 (6) ◽  
pp. 2093-2102 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Efficient Algorithm ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Time Dependent ◽  
Integrable Equations ◽  
Content Type ◽  
Kdv Equations ◽  
Propagation Of Waves ◽  
Fifth Order ◽  
Multiple Complex Soliton Solutions

Purpose The purpose of this paper is concerned with developing two integrable Korteweg de-Vries (KdV) equations of third- and fifth-orders; each possesses time-dependent coefficients. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for these two equations. Design/methodology/approach The integrability of each of the developed models has been confirmed by using the Painlev´e analysis. The author uses the complex forms of the simplified Hirota’s method to obtain two fundamentally different sets of solutions, multiple real and multiple complex soliton solutions for each model. Findings The time-dependent KdV equations feature interesting results in propagation of waves and fluid flow. Research limitations/implications The paper presents a new efficient algorithm for constructing time-dependent integrable equations. Practical implications The author develops two time-dependent integrable KdV equations of third- and fifth-order. These models represent more specific data than the constant equations. The author showed that integrable equation gives real and complex soliton solutions. Social implications The work presents useful findings in the propagation of waves. Originality/value The paper presents a new efficient algorithm for constructing time-dependent integrable equations.

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.

scholarly journals AKNS Formalism and Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients

2016 ◽  
Vol 6 ◽  
pp. 32-41 ◽  
Author(s):  
Supratim Das ◽  
Dibyendu Ghosh
Keyword(s):  
Exact Solutions ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Variable Coefficients ◽  
Jacobian Elliptic Functions ◽  
Generalized Kdv Equation ◽  
New Exact Solutions ◽  
Kdv Equations ◽  
Modified Kdv Equation ◽  
Modified Kdv Equations

We apply the AKNS hierarchy to derive the generalized KdV equation andthe generalized modified KdV equation with variable-coefficients. We system-atically derive new exact solutions for them. The solutions turn out to beexpressible in terms of doubly-periodic Jacobian elliptic functions.

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