Periodic Traveling Wave Solutions Research Articles

Nonlinear stability of periodic-wave solutions for systems of dispersive equations

We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.

Odd periodic waves and stability results for the defocusing mass‐critical Korteweg‐de Vries equation

In this paper, we present results of existence and stability of odd periodic traveling wave solutions for the defocusing mass‐critical Korteweg‐de Vries equation. The existence of periodic wave trains is obtained by solving a constrained minimization problem. Concerning the stability, we use the Floquet theory to determine the behavior of the first three eigenvalues of the linearized operator around the wave, as well as the positiveness of the associated Hessian matrix.

The entry-exit theorem and relaxation oscillations in slow-fast planar systems

The entry-exit theorem for the phenomenon of delay of stability loss for certain types of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. The general existing proofs of this theorem depend on Fenichel's geometric singular perturbation theory and blow-up techniques. In this work, we give a short and elementary proof of the entry-exit theorem based on a direct study of asymptotic formulas of the underlying solutions. We employ this theorem to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of relaxation oscillations. The results are then applied to a diffusive predator-prey model with Holling type II functional response to establish periodic traveling wave solutions. Furthermore, we extend our work to another class of slow-fast systems that can have multiple orbits exhibiting relaxation oscillations, and subsequently apply the results to a two time-scale Holling-Tanner predator-prey model with Holling type IV functional response. It is generally assumed in the literature that the non-trivial equilibrium points exist uniquely in the interior of the domains bounded by the relaxation oscillations; we do not make this assumption in this paper.

Comment on “the solitons and periodic travelling wave solutions for Dodd-Bullough-Mikhailov and Tzitzeica-Dodd-Bullough equations in quantum field theory, Optik 168 (2018) 807-816”

In the recently published article, S.Saha Ray investigated Dodd-Bullough-Mikhailov (DBM) and Tzitzeica-Dodd-Bullough (TDB) equations via his new extended auxiliary equation method. And he claimed that new types of Jacobi elliptic function solutions had been obtained and meanwhile a new family of explicit travelling wave solutions were derived. In this comment, we show that all of exact traveling solutions presented by S.Saha Ray do not satisfy the new extended auxiliary equation method and therefore are incorrect. In addition, general traveling wave solutions of the two differential equation are presented by making full use of certain results published previously.

Critical periodic traveling waves for a periodic and diffusive epidemic model

ABSTRACT This work is devoted to the existence of time periodic traveling wave solutions with critical speed for a seasonal and diffusive epidemic model. We adopt the approach of super- and sub-solutions and Schauder's fixed point theorem to the truncated problem combined with limiting arguments to obtain the existence of critical periodic traveling waves. This would be appropriate for the modification versions of current model, where standard incidence is replaced by other nonlinear incidence.

Time periodic traveling wave solutions for a Kermack–McKendrick epidemic model with diffusion and seasonality

In this paper, we study the time periodic traveling wave solutions for a Kermack–McKendrick SIR epidemic model with individuals diffusion and environment heterogeneity. In terms of the basic reproduction number $$R_0$$ of the corresponding periodic ordinary differential model and the minimal wave speed $$c^*$$ , we establish the existence of periodic traveling wave solutions by the method of super- and sub-solutions, the fixed-point theorem, as applied to a truncated problem on a large but finite interval, and the limiting arguments. We further obtain the nonexistence of periodic traveling wave solutions for two cases involved with $$R_0$$ and $$c^*$$ .

Nonlinear dispersive Alfvén waves interaction in magnetized plasma

This study is concerned with the nonlinear interactions between pairs of intersecting Alfvén waves in a magnetized plasma and used the modified Korteweg–de Vries equation to study nonlinear interactions. The modulation instability analysis shows the existence of periodic traveling wave solution in the system. Two different types of waves interaction solutions, namely, the periodic wave interaction solutions and the solitary wave interaction ones, are captured analytically. It is found that the wave resonance for the periodic waves interaction could happen as various wave numbers are nearly the same. In this case, the subsidiary waves could not be neglected. It is also found that the interaction for solitary waves, different solitons eventually regain their original states. The solitons with higher energy possess more speed as compared to the low energy solitons. The phenomenon of Alfvén wave interaction can be of importance for understanding the transport mechanism of magnetic waves in various processes of heating and transport of energy in space, solar wind, and astrophysical plasma.

Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats

This paper is concerned with propagation phenomena of a general class of partially degenerate nonlocal dispersal cooperative systems in time and space periodic habitats. We first show that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions. Next, we prove that if the wave speed is greater than the spreading speed, there exists a time and space periodic traveling wave solution connecting the stable positive time and space periodic steady state and 0. Finally, we apply these results to two species partially degenerate competition systems and a partially degenerate epidemic model with nonlocal dispersal in time and space periodic habitats.

Stability of periodic waves in Hamiltonian PDEs of either long wavelength or small amplitude

Stability criteria have been derived and investigated in the last decades for many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They turned out to depend in a crucial way on the negative signature of the Hessian matrix of action integrals associated with those waves. In a previous paper (Nonlinearity 2016), the authors addressed the characterization of stability of periodic waves for a rather large class of Hamiltonian partial differential equations that includes quasilinear generalizations of the Korteweg--de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. They derived a sufficient condition for orbital stability with respect to co-periodic perturbations, and a necessary condition for spectral stability, both in terms of the negative signature - or Morse index - of the Hessian matrix of the action integral. Here the asymptotic behavior of this matrix is investigated in two asymptotic regimes, namely for small amplitude waves and for waves approaching a solitary wave as their wavelength goes to infinity. The special structure of the matrices involved in the expansions makes possible to actually compute the negative signature of the action Hessian both in the harmonic limit and in the soliton limit. As a consequence, it is found that nondegenerate small amplitude waves are orbitally stable with respect to co-periodic perturbations in this framework. For waves of long wavelength, the negative signature of the action Hessian is found to be exactly governed by the second derivative with respect to the wave speed of the Boussinesq momentum associated with the limiting solitary wave.

Propagation dynamics of a time periodic and delayed reaction-diffusion model without quasi-monotonicity

In this paper, we consider a time periodic non-monotone and nonlocal delayed reaction-diffusion population model with stage structure. We first prove the existence of the asymptotic speed c ∗ c^* of spread by virtue of two auxiliary equations and comparison arguments. By the method of super- and sub-solutions and the fixed point theorem, as applied to the truncated problem on a finite interval, and the limiting arguments, we then establish the existence of time periodic traveling wave solutions of the model system with wave speed c > c ∗ c>c^* . We further use the results of the asymptotic speed of spread to obtain the non-existence of traveling wave solutions for wave speed c > c ∗ c>c^* . Finally, we prove the existence of the critical periodic traveling wave with wave speed c = c ∗ c=c^* . It turns out that the asymptotic speed of spread coincides with the minimal wave speed for positive periodic traveling waves. These results are also applied to the model system with two prototypical birth functions.

Orbital stability of one-parameter periodic traveling waves for dispersive equations and applications

This paper establishes sufficient conditions for the orbital stability of one-parameter spatially periodic traveling-wave solutions for one-dimensional dispersive equations. Our method of proof combines known techniques with some new ideas. As a consequence of our result, we give several applications for well known dispersive equations. Extension of the theory to regularized equations is also established.

Orbital stability of periodic traveling-wave solutions for a dispersive equation

In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic dnoidal waves associated with the Kawahara equation.

Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation

We study the periodic traveling wave solutions of the derivative nonlinear Schrödinger equation (DNLS). It is known that DNLS has two types of solitons on the whole line; one has exponential decay and the other has algebraic decay. The latter corresponds to the soliton for the massless case. In the new global results recently obtained by Fukaya, Hayashi and Inui [15], the properties of two-parameter of the solitons are essentially used in the proof, and especially the soliton for the massless case plays an important role. To investigate further properties of the solitons, we construct exact periodic traveling wave solutions which yield the solitons on the whole line including the massless case in the long-period limit. Moreover, we study the regularity of the convergence of these exact solutions in the long-period limit. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation.

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 Lamé 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.

Nonlinear travelling internal waves with piecewise-linear shear profiles

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model

We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of $${2\pi}$$ -periodic traveling wave solutions continuing from the zero state. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like |x|1/2 near its crest, the cusped waves obtained here behave like |x log |x||. Although the linear operator involved in this equation can be easily represented in terms of an integral operator, it maps continuous functions out of the Holder and Lipschitz scales of function spaces by introducing logarithmic singularities. Since the nonlinearity is also of higher order than that of the unidirectional Whitham equation, several parts of our proofs and results deviate from those of the corresponding unidirectional equation, with the analysis of the logarithmic singularity being the most subtle component. This paper is part of a longer research programme for understanding the interplay between nonlinearities and dispersion in the formation of large-amplitude waves and their singularities.

On the instability of elliptic traveling wave solutions of the modified Camassa–Holm equation

This paper is concerned with the orbital instability for a specific class of periodic traveling wave solutions with the mean zero property and large spatial period related to the modified Camassa–Holm equation. These solutions, called snoidal waves, are written in terms of the Jacobi elliptic functions. To prove our result we use the abstract method of Grillakis, Shatah and Strauss [23], the Floquet theory for periodic eigenvalue problems and the n-gaps potentials theory of Dubrovin, Matveev and Novikov [19].

Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems

The present paper is concerned with the spatial spreading speeds and traveling wave solutions of cooperative systems in space-time periodic habitats with nonlocal dispersal. It is assumed that the trivial solution ${\bf u} = {\bf 0}$ of such a system is unstable and the system has a stable space-time periodic positive solution ${\bf u^*}(t,x)$. We first show that in any direction $ξ∈ \mathbb{S}^{N-1}$, such a system has a finite spreading speed interval, and under certain condition, the spreading speed interval is a singleton set, and hence, the system has a single spreading speed $c^{*}(ξ)$ in the direction of $ξ$. Next, we show that for any $c>c^{*}(ξ)$, there are space-time periodic traveling wave solutions of the form ${\bf{u}}(t,x) = {\bf{Φ}}(x-ctξ,t,ctξ)$ connecting ${\bf u^*}$ and ${\bf 0}$, and propagating in the direction of $ξ$ with speed $c$, where $Φ(x,t,y)$ is periodic in $t$ and $y$, and there is no such solution for $c<c^{*}(ξ)$. We also prove the continuity and uniqueness of space-time periodic traveling wave solutions when the reaction term is strictly sub-homogeneous. Finally, we apply the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity.

Periodic traveling waves in a reaction–diffusion model with chemotaxis and nonlocal delay effect

In this paper, we study the existence of periodic traveling wave solutions with large wave speed for a reaction–diffusion model with chemotaxis and nonlocal delay effect by applying the perturbation method. The proof relies on an abstract formulation of the wave profile as a solution of an operator equation in a certain Banach space, coupled with the Lyapunov–Schmidt reduction and the implicit function theorem.

Reduction to modified KdV and its KP-like generalization via phase modulation

The main observation of this paper is that the modified Korteweg–de Vries equation has its natural origin in phase modulation of a basic state such as a periodic travelling wave, or more generally, a family of relative equilibria. Extension to 2 + 1 suggests that a modified Kadomtsev–Petviashvili (or a Konopelchenko–Dubrovsky) equation should emerge, but our result shows that there is an additional term which has gone heretofore unnoticed. Thus, through the novel application of phase modulation a new equation appears as the 2 + 1 extension to a previously known one. To demonstrate the theory it is applied to the cubic-quintic nonlinear Schrödinger (CQNLS) equation, showing that there are relevant parameter values where a modified KP equation bifurcates from periodic travelling wave solutions of the 2 + 1 CQNLS equation.