scholarly journals The New Extended KdV Equation for the Case of an Uneven Bottom

2018 ◽  
Vol 24 (4) ◽  
pp. 221-225
Author(s):  
Piotr Rozmej ◽  
Anna Karczewska
Keyword(s):  
Kdv Equation ◽  
Uneven Bottom ◽  
Extended Kdv Equation

scholarly journals A Theoretical Study of an Extended KDV Equation

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Explicit Solution ◽  
Kdv Equation ◽  
Integrable Equation ◽  
Weakly Nonlinear ◽  
Kdv Equations ◽  
Weakly Nonlinear Waves ◽  
Extended Kdv Equation ◽  
The One ◽  
Definition Of ◽  
Fifth Order

Discovered experimentally by Russell and described theoretically by Korteweg and de Vries, KdV equation has been a nonlinear evolution equation describing the propagation of weakly dispersive and weakly nonlinear waves. This equation received a lot of attention from mathematical and physical communities as an integrable equation. The objectives of this paper are: first, providing a rigorous mathematical derivation of an extended KdV equations, one on the velocity, other on the surface elevation, next, solving explicitly the one on the velocity. In order to derive rigorously these equations, we will refer to the definition of consistency, and to find an explicit solution for this equation, we will use the sine-cosine method. As a result of this work, a rigorous justification of the extended Kdv equation of fifth order will be done, and an explicit solution of this equation will be derived.

scholarly journals Exact Cnoidal Solutions of the Extended KdV Equation

2018 ◽  
Vol 133 (5) ◽  
pp. 1191-1199 ◽  
Author(s):  
E. Infeld ◽  
A. Karczewska ◽  
G. Rowlands ◽  
P. Rozmej
Keyword(s):  
Kdv Equation ◽  
Extended Kdv Equation

On some localized waves described by the extended KdV equation

2005 ◽  
Vol 333 (7) ◽  
pp. 528-533 ◽  
Author(s):  
Alexey V. Porubov ◽  
Gérard A. Maugin ◽  
Vitaly V. Gursky ◽  
Valeria V. Krzhizhanovskaya
Keyword(s):  
Kdv Equation ◽  
Extended Kdv Equation ◽  
Localized Waves

scholarly journals Single soliton solution to the extended KdV equation over uneven depth

2017 ◽  
Vol 40 (11) ◽  
Author(s):  
George Rowlands ◽  
Piotr Rozmej ◽  
Eryk Infeld ◽  
Anna Karczewska
Keyword(s):  
Soliton Solution ◽  
Kdv Equation ◽  
Extended Kdv Equation

Nonlinear excitations and dynamic features of dust ion-acoustic waves in a magnetized electron–positron–ion plasma

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Rabindranath Maity ◽  
Biswajit Sahu
Keyword(s):  
Numerical Simulation ◽  
Acoustic Waves ◽  
Kdv Equation ◽  
Phase Portraits ◽  
Dust Particles ◽  
Superthermal Electrons ◽  
Ion Acoustic Waves ◽  
Nonlinear Excitations ◽  
Ion Acoustic ◽  
Extended Kdv Equation

Abstract A wide class of nonlinear excitations and the dynamics of wave groups of finite amplitude ion-acoustic waves are investigated in multicomponent magnetized plasma system comprising warm ions, and superthermal electrons as well as positrons in presence of negatively charged impurities or dust particles. Employing the reductive perturbation technique (RPT), the Korteweg–de-Vries (KdV) equation, and extended KdV equation are derived. The presence of excess superthermal electrons as well as positrons and other plasma parameters are shown to influence the characteristics of both compressive and rarefactive solitons as well as double layers (DLs). Also, we extend our investigation by deriving the nonlinear Schrödinger equation from the extended KdV equation employing a suitable transformation to study the wave group dynamics for long waves. The analytical and numerical simulation results demonstrate that nonlinear wave predicts solitons, “table-top” solitons, DLs, bipolar structure, rogue waves, and breather structures. Moreover, implementing the concept of dynamical systems, phase portraits of nonlinear periodic, homoclinic trajectories, and supernonlinear periodic trajectories are presented through numerical simulation.

scholarly journals Superposition solutions to the extended KdV equation for water surface waves

2017 ◽  
Vol 91 (2) ◽  
pp. 1085-1093 ◽  
Author(s):  
Piotr Rozmej ◽  
Anna Karczewska ◽  
Eryk Infeld
Keyword(s):  
Surface Waves ◽  
Water Surface ◽  
Kdv Equation ◽  
Extended Kdv Equation

scholarly journals A rigorous derivation of the extended KdV equation

2020 ◽  
Vol 1564 ◽  
pp. 012006
Author(s):  
M Berjawi ◽  
T ElArwadi ◽  
S Israwi
Keyword(s):  
Kdv Equation ◽  
Rigorous Derivation ◽  
Extended Kdv Equation
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