Classification of Traveling Waves for a Class of Nonlinear Wave Equations

2006 ◽  
Vol 18 (2) ◽  
pp. 381-391 ◽  
Author(s):  
Jonatan Lenells
Keyword(s):  
Nonlinear Wave ◽  
Traveling Waves ◽  
Wave Equations ◽  
Nonlinear Wave Equations

EXACT SOLUTIONS AND THEIR DYNAMICS OF TRAVELING WAVES IN THREE TYPICAL NONLINEAR WAVE EQUATIONS

2009 ◽  
Vol 19 (07) ◽  
pp. 2249-2266 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Nonlinear Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Wave Equations ◽  
Visual Illusion ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Dynamical Analysis ◽  
Nonlinear Wave Equations ◽  
Wave Solutions

It was reported in the literature that some nonlinear wave equations have the so-called loop- and inverted-loop-soliton solutions, as well as the so-called loop-periodic solutions. Are these true mathematical solutions or just numerical artifacts? To answer the question, this article investigates all traveling wave solutions in the parameter space for three typical nonlinear wave equations from a theoretical viewpoint of dynamical systems. Dynamical analysis shows that all these loop- and inverted-loop-solutions are merely visual illusion of numerical artifacts. To reveal the nature of such special phenomena, this article also offers the mathematical parametric representations of these traveling wave solutions precisely in analytic forms.

Group classification of third order nonlinear wave equations chain’s

2020 ◽  
Vol 15 (3-4) ◽  
pp. 232-237
Author(s):  
O.K. Babkov ◽  
G.Z. Mukhametova
Keyword(s):  
Lie Algebras ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Group Analysis ◽  
Group Classification ◽  
Nonlinear Wave Equations ◽  
Bäcklund Transformations ◽  
Third Order ◽  
A Chain

The paper presents the results of point symmetrys Lie algebras for third-order nonlinear wave equations calculating linked into a chain by Bäcklund transformations. Calculations are carried out by using Lie-Ovsyannikov method of group analysis. The basic algebras of point symmetries of the indicated equations are found, all possible cases of their extension are revealed, and the commutator tables of algebras found are calculated.

scholarly journals Complete group classification of a class of nonlinear wave equations

2012 ◽  
Vol 53 (12) ◽  
pp. 123515 ◽  
Author(s):  
Alexander Bihlo ◽  
Elsa Dos Santos Cardoso-Bihlo ◽  
Roman O. Popovych
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Group Classification ◽  
Nonlinear Wave Equations ◽  
Complete Group

Classification of all travelling-wave solutions for some nonlinear dispersive equations

2007 ◽  
Vol 365 (1858) ◽  
pp. 2291-2298 ◽  
Author(s):  
Jonatan Lenells
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Travelling Wave ◽  
Nonlinear Wave Equations ◽  
Dispersive Equations ◽  
Travelling Wave Solutions ◽  
Nonlinear Dispersive Equations ◽  
Wave Solutions ◽  
Wave Profiles

We present a method for the classification of all weak travelling-wave solutions for some dispersive nonlinear wave equations. When applied to the Camassa–Holm or the Degasperis–Procesi equation, the approach shows the existence of not only smooth, peaked and cusped travelling-wave solutions, but also more exotic solutions with fractal-like wave profiles.

Sign in / Sign up

Export Citation Format

Share Document