scholarly journals Quasiperiodic Solutions of Completely Resonant Wave Equations with Quasiperiodic Forced Terms

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Yixian Gao ◽  
Weipeng Zhang ◽  
Jing Chang
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Traveling Waves ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Linking Theorem ◽  
First Order ◽  
Quasiperiodic Solutions ◽  
Resonant Wave ◽  
Spatial Boundary

This paper is concerned with the existence of quasiperiodic solutions with two frequencies of completely resonant, quasiperiodically forced nonlinear wave equations subject to periodic spatial boundary conditions. The solutions turn out to be, at the first order, the superposition of traveling waves, traveling in the opposite or the same directions. The proofs are based on the variational Lyapunov-Schmidt reduction and the linking theorem, while the bifurcation equations are solved by variational methods.

Characteristics of solitary waves, quasiperiodic solutions, homoclinic breather solutions and rogue waves in the generalized variable-coefficient forced Kadomtsev–Petviashvili equation

2017 ◽  
Vol 31 (36) ◽  
pp. 1750350 ◽  
Author(s):  
Xue-Wei Yan ◽  
Shou-Fu Tian ◽  
Min-Jie Dong ◽  
Li Zou
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Variable Coefficient ◽  
Wave Equations ◽  
Direct Method ◽  
Dynamical Behavior ◽  
Rogue Waves ◽  
Nonlinear Wave Equations ◽  
Quasiperiodic Solutions ◽  
Petviashvili Equation

In this paper, the generalized variable-coefficient forced Kadomtsev–Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

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.

scholarly journals Exact Solution of Linear and Nonlinear Wave Equations by Double Laplace and Iterative Method

2019 ◽  
Vol 11 (5) ◽  
pp. 33
Author(s):  
Adesina K. Adio ◽  
Wumi S. Ajayi ◽  
Olabode O. Bamisile ◽  
Babatunde T. Akanbi
Keyword(s):  
Exact Solution ◽  
Iterative Method ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Iteration Process ◽  
Nonlinear Wave Equations ◽  
Double Laplace Transform ◽  
Linear And Nonlinear

A coupling of double Laplace transform with Iterative method is used to solve linear and nonlinear wave equations subject to initial and boundary conditions. The iteration process leads to disappearance of noise terms and exact solution is obtained at first iteration. Through several examples, the convenience and efficiency of the method is demonstrated, showing its usefulness to overcome difficulties associated with some existing techniques.

scholarly journals Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation

2003 ◽  
Vol 2003 (18) ◽  
pp. 1111-1136 ◽  
Author(s):  
Xiaoping Yuan
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Invariant Manifold ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Invariant Tori ◽  
Nonlinear Wave Equations ◽  
Elliptic Type ◽  
Quasiperiodic Solutions

It is shown that there are plenty of hyperbolic-elliptic invariant tori, thus quasiperiodic solutions for a class of nonlinear wave equations.

ON BOUNDARY CONDITIONS FOR FIRST-ORDER SYMMETRIC HYPERBOLIC SYSTEMS WITH CONSTRAINTS

2013 ◽  
Vol 10 (04) ◽  
pp. 725-734 ◽  
Author(s):  
NICOLAE TARFULEA
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
The Cauchy Problem ◽  
Initial Boundary Value ◽  
Well Posed

The Cauchy problem for many first-order symmetric hyperbolic (FOSH) systems is constraint preserving, i.e. the solution satisfies certain spatial differential constraints whenever the initial data does. Frequently, artificial space cut-offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artificial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal non-negative boundary conditions. Based on a characterization of maximal non-negative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by analyzing a system of wave equations in a first-order formulation subject to divergence constraints.

Sign in / Sign up

Export Citation Format

Share Document