A study on a two-wave mode Kadomtsev-Petviashvili equation: conditions for multiple soliton solutions to exist

2017 ◽  
Vol 40 (11) ◽  
pp. 4128-4133 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Wave Mode ◽  
Multiple Soliton Solutions ◽  
Petviashvili Equation

Two integrable extensions of the Kadomtsev-Petviashvili equation

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Kp Equation ◽  
Completely Integrable ◽  
Multiple Soliton Solutions ◽  
Petviashvili Equation ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

AbstractIn this work, two new completely integrable extensions of the Kadomtsev-Petviashvili (eKP) equation are developed. Multiple soliton solutions and multiple singular soliton solutions are derived to demonstrate the compatibility of the extensions of the KP equation.

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.

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.

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Shallow Water Wave Equations

Purpose This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space. Design/methodology/approach The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense. Findings The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium. Research limitations/implications The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method. Practical implications The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition. Social implications The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions. Originality/value The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

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