scholarly journals Existence and Orbital Stability of Cnoidal Waves for a 1D Boussinesq Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-36 ◽  
Author(s):  
Jaime Angulo ◽  
Jose R. Quintero
Keyword(s):  
Orbital Stability ◽  
Type Equation ◽  
Elliptic Functions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Jacobian Elliptic Functions ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Complete Study ◽  
Set Up

We will study the existence and stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equationΦtt−Φxx+aΦxxxx−bΦxxtt+ΦtΦxx+2ΦxΦxt=0. Periodic travelling-wave solutions with an arbitrary fundamental periodT0will be built by using Jacobian elliptic functions. Stability (orbital) of these solutions by periodic disturbances with periodT0will be a consequence of the general stability criteria given by M. Grillakis, J. Shatah, and W. Strauss. A complete study of the periodic eigenvalue problem associated to the Lame equation is set up.

scholarly journals Exact Solutions of Travelling Wave Model via Dynamical System Method

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Heng Wang ◽  
Longwei Chen ◽  
Hongjiang Liu
Keyword(s):  
Dynamical System ◽  
Solitary Wave ◽  
Elliptic Functions ◽  
Wave Model ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Jacobian Elliptic Functions ◽  
Wave Solutions ◽  
Periodic Travelling Wave

By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrödinger-Boussinesq equations are studied. Based on this method, the bounded exact travelling wave solutions are obtained which contain solitary wave solutions and periodic travelling wave solutions. The solitary wave solutions and periodic travelling wave solutions are expressed by the hyperbolic functions and the Jacobian elliptic functions, respectively. The results show that the presented findings improve the related previous conclusions. Furthermore, the numerical simulations of the solitary wave solutions and the periodic travelling wave solutions are given to show the correctness of our results.

scholarly journals A Note on the Stability and Instability of Travelling Wave of Korteweg-de Vries Type: The Periodic Case

2011 ◽  
Vol 14 ◽  
pp. 57-72
Author(s):  
José R Quintero
Keyword(s):  
Orbital Stability ◽  
Type Equation ◽  
Travelling Wave ◽  
Periodic Case ◽  
Travelling Wave Solutions ◽  
Stability And Instability ◽  
Periodic Travelling Wave ◽  
De Vries ◽  
The Mean ◽  
The Stability

In this paper we adapt the work of M. Grillakis, J. Shatah, and W. Strauss, or J. Bona, P. Souganidis and W. Strauss to the periodic case in spaces having the mean zero property in order to establish the orbital stability/instability of periodic travelling wave solutions of a generalized Korteweg-de Vries type equation.

scholarly journals On the Instability of a Class of Periodic Travelling Wave Solutions of the Modified Boussinesq Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-21
Author(s):  
Lynnyngs Kelly Arruda
Keyword(s):  
Boussinesq Equation ◽  
Elliptic Functions ◽  
Travelling Wave ◽  
Energy Space ◽  
Fundamental Period ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Modified Boussinesq Equation

This paper is concerned with instability of periodic travelling wave solutions of the modified Boussinesq equation. Periodic travelling wave solutions with a fixed fundamental periodLwill be constructed by using Jacobi's elliptic functions. It will be shown that these solutions, calleddnoidal waves, are nonlinearly unstable in the energy space for a range of their speeds of propagation and periods.

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.

Travelling And Periodic Wave Solutions Of Some Nonlinear Wave Equations

2004 ◽  
Vol 59 (7-8) ◽  
pp. 389-396 ◽  
Author(s):  
A. H. Khater ◽  
M. M. Hassan
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Nonlinear Wave Equations ◽  
Travelling Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Pacs 02.30

We present the mixed dn-sn method for finding periodic wave solutions of some nonlinear wave equations. Introducing an appropriate transformation, we extend this method to a special type of nonlinear equations and construct their solutions, which are not expressible as polynomials in the Jacobi elliptic functions. The obtained solutions include the well known kink-type and bell-type solutions as a limiting cases. Also, some new travelling wave solutions are found. - PACS: 02.30.Jr; 03.40.Kf

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