scholarly journals Continuation methods for time-periodic travelling-wave solutions to evolution equations

2018 ◽  
Vol 86 ◽  
pp. 291-297 ◽  
Author(s):  
T.-S. Lin ◽  
D. Tseluiko ◽  
M.G. Blyth ◽  
S. Kalliadasis
Keyword(s):  
Evolution Equations ◽  
Travelling Wave ◽  
Continuation Methods ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Time Periodic

scholarly journals Singularity analysis and analytic solutions for the Benney–Gjevik equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andronikos Paliathanasis ◽  
Genly Leon ◽  
P. G. L. Leach
Keyword(s):  
Evolution Equations ◽  
Singularity Analysis ◽  
Analytic Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Painlevé Test ◽  
Wave Solutions ◽  
Algebraic Solutions

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

scholarly journals Nonuniform Continuity of the Osmosis K(2, 2) Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Aiyong Chen ◽  
Yong Ding ◽  
Wentao Huang
Keyword(s):  
Differential Equations ◽  
Small Amplitude ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Qualitative Theory ◽  
Travelling Wave Solutions ◽  
Solution Map ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Uniformly Continuous

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

scholarly journals Periodic Travelling Wave Solutions of Discrete Nonlinear Schrödinger Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Dirk Hennig
Keyword(s):  
Fixed Point ◽  
Fixed Point Problem ◽  
Travelling Wave ◽  
Point Problem ◽  
Travelling Wave Solutions ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Discrete Nonlinear Schrödinger Equations

The existence of nonzero periodic travelling wave solutions for a general discrete nonlinear Schrödinger equation (DNLS) on one-dimensional lattices is proved. The DNLS features a general nonlinear term and variable range of interactions going beyond the usual nearest-neighbour interaction. The problem of the existence of travelling wave solutions is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem.

The Functional Variable Method to Find New Exact Solutions of the Nonlinear Evolution Equations with Dual-Power-Law Nonlinearity

2020 ◽  
Vol 21 (3-4) ◽  
pp. 249-257 ◽  
Author(s):  
Hadi Rezazadeh ◽  
Javad Vahidi ◽  
Asim Zafar ◽  
Ahmet Bekir
Keyword(s):  
Power Law ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Functional Variable ◽  
Functional Variable Method ◽  
Dual Power

AbstractIn this work, we established new travelling wave solutions for some nonlinear evolution equations with dual-power-law nonlinearity namely the Zakharov–Kuznetsov equation, the Benjamin–Bona–Mahony equation and the Korteweg–de Vries equation. The functional variable method was used to construct travelling wave solutions of nonlinear evolution equations with dual-power-law nonlinearity. The travelling wave solutions are expressed by generalized hyperbolic functions and the rational functions. This method presents a wider applicability for handling nonlinear wave equations.

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