Bifurcation and nonsmooth dynamics of solitary waves in the generalized long-short wave equations

Generalized wave equations, which model the resonant interaction between the long wave and the short wave, are considered. To understand the underlying complex dynamics, the bifurcations and nonsmooth behaviors of solitary waves for this system are investigated by qualitative techniques in dynamical systems. These complex behaviors may serve as mechanisms for fascinating physical phenomena such as solitons, chaos and turbulence.     

Stability of traveling waves for Hamilton-Jacobi equations with finite speed perturbations

We prove nonlinear stability of planar shock for general Hamilton-Jacobi equations with finite speed perturbation. Here we use energy estimates. It is shown that the solution connecting a weak shock is asymptotically stable under small perturbations.     

Solitary waves and their linear stability in weakly coupled KdV equations

We consider a system of weakly coupled KdV equations developed initially by Gear & Grimshaw to model interactions between long waves. We prove the existence of a variety of solitary wave solutions, some of which are not constrained minimizers. We show that such solutions are always linearly unstable. Moreover, the nature of the instability may be oscillatory and as such provides a rigorous justification for the numerically observed phenomenon of “leapfrogging.”     

Periodic traveling waves and locating oscillating patterns in multidimensional domains

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in , when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.

     

Orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations $\left\{\begin{array}{l}i\varepsilon_{t}+\varepsilon_{xx}=n\varepsilon+\alpha|\varepsilon|^{2}\varepsilon,\\n_{t}=(|\varepsilon|^{2})_{x}, x\in R.\end{array} \right.$ Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $L$ for the generalized Long-Short wave equations. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period $L$. As the modulus of the Jacobian elliptic function $k\rightarrow 1$, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equations. In particular, as $\alpha=0$, we can also obtain the orbital stability results of periodic wave solutions and solitary wave solutions for the long-short wave resonance equations. The results in the present paper improve and extend the previous stability results of long-shore wave equations and its extension equations.     

Coexistence of the solitary and periodic waves in convecting shallow water fluid

The existence of a solitary wave for the shallow water model in convecting circumstance was established in previous works. It is still unknown that whether there exist periodic waves. In this paper, we prove that the models possess periodic waves with a fixed range of wave speed. The amplitude and wave speed are explicitly given. Moreover, the coexistence of the solitary wave and one periodic wave is established.     

Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation

By using the method of planar dynamical systems to a nonlinear variant of the regularized long-wave equation (RLW equation in short), the existence of smooth and non-smooth solitary wave (so called peakon and valleyon) and infinite many periodic wave solutions is shown. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. The formulas to compute the travelling waves are also educed. We notice that some results in [Wazwaz AM. Analytic study on nonlinear variants of the RLW and the PHI-four equations. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2005.03.001] are incorrect.     

Solitary wave and chaotic behavior of traveling wave solutions for the coupled KdV equations

Using the method of dynamical systems to study the coupled KdV system, some exact explicit parametric representations of the solitary wave and periodic wave solutions are obtained in the given parameter regions. Chaotic behavior of traveling wave solutions is determined.     

Orbital stability of periodic traveling wave solutions to the generalized zakharov equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations

$\{\begin{array}{c}i{u}_t+u_{xx}=uv+|u{|}^{2}u,\\ v_{tt}-v_{xx}=\left(|u{|}^2\right)xx\end{array}.$

First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].     

Stable periodic traveling waves for a predator-prey model with non-constant death rate and delay

In this paper we will consider a predator-prey model with a non-constant death rate and distributed delay, described by a partial integro-differential system. The main goal of this work is to prove that the partial integro-differential system has periodic orbitally asymptotically stable solutions in the form of periodic traveling waves; i.e. N(x, t) = N(σt − μ · x), P(x, t) = P(σt − μ · x), where σ > 0 is the angular frequency and μ is the vector number of the plane wave, which propagates in the direction of the vector μ with speed c = σ/∥μ∥; and N(x, t) and P(x, t) are the spatial population densities of the prey and the predator species, respectively. In order to achieve our goal we will use singular perturbation’s techniques.     

Orbital stability of solitary waves for the Klein-Gordon-Zakharov equations

1.IntroductionInthetheoreticalinvestigationsofthedynamicsofstrongLangmuirturbulenceinplasmaphysics,varioustypesofZakharovequationstakeanimportantrole(see[3--8]).Intillspaper,weconsiderthefollowingKlein-Gordon-Zakharovequations:{:ti:::::;.;;:~"aiR,(11)withuacomplexfunctionandnarealfunction.Thelocalandglobalekistenceoftheinitialvalueproblemfor(1.l)wasconsideredin[4,6].Inthispaper,weconsidertheorbitalstabilityofthesolitarywavesof(1.l).Byapplyingtheabstracttheoryof[1,2]anddetailedspectralanalys…     

Existence and uniqueness of traveling waves for non-monotone integral equations with application

This paper is concerned with the traveling waves in a class of non-monotone integral equations. First we establish the existence of traveling waves. The approach is based on the construction of two associated auxiliary monotone integral equations and a profile set in a suitable Banach space. Then we show that the traveling waves are unique up to translations under some reasonable assumptions. The exact asymptotic behavior of the profiles as ξ→−∞ and the existence of minimal wave speed are also obtained. Finally, we apply our results to an epidemic model with non-monotone “force of infection”.     

A normalized solitary wave solution of the Maxwell-Dirac equations

We prove the existence of a $L^2$-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.     

Stability of periodic traveling waves for complex modified Korteweg-de Vries equation

We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.     

The stability of traveling waves with non-critical speeds for some competition systems

This paper is concerned with some quasilinear cross-diffusion systems which model competing species in mathematical ecology.By detailed spectral analysis,each traveling wave solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted spaces.     

Qualitative Analysis and Periodic Cusp Waves to a Class of Generalized Short Pulse Equations

In this paper, we qualitatively study periodic cusp waves to a class of generalized short pulse equations, which are of the general form of three special generalized short pulse equations, from the perspective of dynamical systems. We show the existence of smooth periodic waves, periodic cusp wave and compactons, obtain exact expression of periodic cusp wave and illustrate the limiting process of periodic cusp wave from smooth periodic waves.     

Periodic traveling waves of a mean curvature equation in high dimensional cylinders

Let Ω be the unit ball in R^N. Consider the mean curvature equation

(E0)     

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction-diffusion equations

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling waves of systems of reaction-diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider?s renormalization techniques do not appear to apply.     

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

We present a general result of transverse nonlinear instability of 1d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1-d model and the transverse perturbation “have the same sign”. Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Boussinesq system and the Zakharov–Kuznetsov equation and we hope that it may be useful in other contexts.