Multiple complex soliton solutions for integrable negative-order KdV and integrable negative-order modified KdV equations

2019 ◽  
Vol 88 ◽  
pp. 1-7 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Kdv Equations ◽  
Negative Order ◽  
Modified Kdv Equations ◽  
Multiple Complex Soliton Solutions

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.

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.

Two wave mode higher-order modified KdV equations

2017 ◽  
Vol 27 (10) ◽  
pp. 2223-2230 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Wave Model ◽  
Wave Mode ◽  
Higher Order ◽  
Dispersion Parameters ◽  
Content Type ◽  
Kdv Equations ◽  
Multiple Soliton Solutions ◽  
Modified Kdv Equations ◽  
Propagation Of Waves

Purpose The purpose of this paper is concerned with developing two-mode higher-order modified Korteweg-de Vries (KdV) equations. The study shows that multiple soliton solutions exist for essential conditions related to the nonlinearity and dispersion parameters. Design/methodology/approach The proposed technique for constructing a two-wave model, as presented in this work, has been shown to be very efficient. The employed approach formally derives the essential conditions for soliton solutions to exist. Findings The examined two-wave model features interesting results in propagation of waves and fluid flow. Research limitations/implications The paper presents a new and efficient algorithm for constructing and studying two-wave-mode higher-order modified KdV equations. Practical implications A two-wave model was constructed for higher-order modified KdV equations. The essential conditions for multiple soliton solutions to exist were derived. Social implications The work shows the distinct features of the standard equation and the newly developed equation. Originality/value The work is original and this is the first time for two-wave-mode higher-order modified KdV equations to be constructed and studied.

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