On the traveling waves governed by the Camassa–Holm equation

The Camassa–Holm equation admits undistorted traveling waves that are either smooth or exhibit peaks or cusps. All three wave types can be periodic or solitary. Also waves of different types may be combined. In the present paper it is shown that, apart from peaks and cusps, the traveling waves governed by the Camassa–Holm equation can be found from some simpler equation. In the case of peaked solutions, this reduced equation is even linear. The governing equation of traveling waves in its original form can be interpreted as a nonlinear combination of the reduced equation and its first integral. For a small range of the integration constant, the reduced equation admits bounded solutions, which then are directly inherited by the Camassa–Holm equation. In general, the solutions of the reduced equation are unbounded and cannot be considered to represent traveling waves. The full equation, however, has a nonlinearity in the highest derivative, which is characteristic for the Camassa–Holm and some other equations. This nonlinear term offers the possibility of constructing bounded traveling waves from the unbounded solutions of the reduced equation. These waves necessarily have discontinuities in the slope and are, therefore, solutions only in a generalized sense.     

Traveling Waves in Discrete Periodic Media for Bistable Dynamics

This paper is concerned with the existence, uniqueness, and global stability of traveling waves in discrete periodic media for a system of ordinary differential equations exhibiting bistable dynamics. The main tools used to prove the uniqueness and asymptotic stability of traveling waves are the comparison principle, spectrum analysis, and constructions of super/subsolutions. To prove the existence of traveling waves, the system is converted to an integral equation which is common in the study of monostable dynamics but quite rare in the study of bistable dynamics. The main purpose of this paper is to introduce a general framework for the study of traveling waves in discrete periodic media.     

Time Periodic Traveling Waves for a Periodic and Diffusive SIR Epidemic Model

In this paper, we study the time periodic traveling wave solutions for a periodic SIR epidemic model with diffusion and standard incidence. We establish the existence of periodic traveling waves by investigating the fixed points of a nonlinear operator defined on an appropriate set of periodic functions. Then we prove the nonexistence of periodic traveling via the comparison arguments combined with the properties of the spreading speed of an associated subsystem.     

Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity

We prove the existence of multidimensional traveling wave solutions of the bistable reaction-diffusion equation with periodic coefficients under the condition that these coefficients are close to constants. In the case of one space dimension, we prove their asymptotic stability.     

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov＇s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.     

Traveling Waves in Spatial SIRS Models

We study traveling wavefront solutions for two reaction–diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle’s invariance principle, and their uniqueness by a geometric singular perturbation argument.     

Exact solutions,conservation laws,bifurcation of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver equation

In the present work, we observe the dynamical behavior of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver (STO) equation. Exact solutions are derived using \({1}/{G^{^{\prime }}}\) expansion and modified Kudryashov methods. The wave transformation is used to transform STO equation into an ordinary differential equation. Combining Runge–Kutta fourth-order and Fourier spectral technique, we use a mixed scheme for the numerical study of STO equation. Since spectral methods expand the solution in trigonometric series resulting into higher-order technique and Runge–Kutta produces improved accuracy, we extract these qualities for a mixed scheme. Results so produced are presented graphically which provide a useful information about the dynamical behavior. Bifurcation behavior of nonlinear and supernonlinear traveling waves of STO equation is studied with the help of bifurcation theory of planar dynamical systems. It is observed that STO equation supports nonlinear solitary wave, periodic wave, shock wave, stable oscillatory wave and most important supernonlinear periodic wave.     

Existence,Uniqueness, and Stability of Generalized Traveling Waves in Time Dependent Monostable Equations

The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic, it is almost periodic in the sense that its wave profile and wave speed are almost periodic.     

Generalized Traveling Waves in Disordered Media: Existence, Uniqueness, and Stability

We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translations of the front. In the case of stationary ergodic reactions, the fronts are proved to propagate with a deterministic positive speed. Our results extend to reaction-advection-diffusion equations with periodic advection and diffusion.     

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes a high-order finite-element method with smooth, periodic splines in space and explicit Runge–Kutta methods in time. Other forms of solutions such as cnoidal waves and dispersive shock waves are also considered. The differences between solutions of the Serre equations and the Euler equations are also studied.     

The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.     

Bifurcations and exact solutions of an asymptotic rotation-Camassa–Holm equation

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth’s rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained.

     

Combustion Fronts in a Porous Medium with Two Layers

We study a model for the lateral propagation of a combustion front through a porous medium with two parallel layers having different properties. The reaction involves oxygen and a solid fuel. In each layer, the model consists of a nonlinear reaction–diffusion–convection system, derived from balance equations and Darcy’s law. Under an incompressibility assumption, we obtain a simple model whose variables are temperature and unburned fuel concentration in each layer. The model includes heat transfer between the layers. We find a family of traveling wave solutions, depending on the heat transfer coefficient and other system parameters, that connect a burned state behind the combustion front to an unburned state ahead of it. These traveling waves are strong: they correspond to connecting orbits of a system of five ordinary differential equations that lie in the unstable manifold of a hyperbolic saddle and the stable manifold of a nonhyperbolic equilibrium. We argue that for physically relevant initial conditions, traveling waves that correspond to connecting orbits that approach the nonhyperbolic equilibrium along its center direction do not occur. When the heat transfer coefficient is small, we prove that strong traveling waves exist for a small range of system parameters, near parameter values where the two layers individually admit strong traveling waves with the same speed. When the heat transfer coefficient is large, we prove that strong traveling waves exist for a very large range of parameters. For small heat transfer, combustion typically does not occur simultaneously in the two layers; for large heat transfer, it does. The proofs use geometric singular perturbation theory. We give a numerical method to solve the nonlinear problem, and we present numerical simulations that indicate that the traveling waves we have found are in fact the dominant feature of solutions.     

Uniqueness of the Bistable Traveling Wave for Mutualist Species

For a system of reaction–diffusion equations that models the interaction of n mutualist species, the existence of the bistable traveling wave solution has been proved where the nonlinear reaction terms possess a certain type of monotonicity. However the problem of whether there can be two distinct traveling waves remains open. In this paper we use a homotopy approach incorporated with the Liapunov–Schmidt method to show that the bistable traveling wave solution is unique. Our method developed in this paper can also be applied to study the existence and uniqueness of traveling wave solutions for some competition models.     

New explicit and exact traveling wave solutions for two nonlinear evolution equations

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Traveling Waves in Diffusive Random Media

The current paper is devoted to the study of traveling waves in diffusive random media, including time and/or space recurrent, almost periodic, quasiperiodic, periodic ones as special cases. It first introduces a notion of traveling waves in general random media, which is a natural extension of the classical notion of traveling waves. Roughly speaking, a solution to a diffusive random equation is a traveling wave solution if both its propagating profile and its propagating speed are random variables. Then by adopting such a point of view that traveling wave solutions are limits of certain wave-like solutions, a general existence theory of traveling waves is established. It shows that the existence of a wave-like solution implies the existence of a critical traveling wave solution, which is the traveling wave solution with minimal propagating speed in many cases. When the media is ergodic, some deterministic \hbox{properties} of average propagating profile and average propagating speed of a traveling wave solution are derived. When the media is compact, certain continuity of the propagating profile of a critical traveling wave solution is obtained. Moreover, if the media is almost periodic, then a critical traveling wave solution is almost automorphic and if the media is periodic, then so is a critical traveling wave solution. Applications of the general theory to a bistable media are discussed. The results obtained in the paper generalize many existing ones on traveling waves. AMS Subject Classification: 35K55, 35K57, 35B50     

Entire Solutions with Annihilating Fronts to a Nonlocal Dispersal Equation with Bistable Nonlinearity and Spatio-Temporal Delay

This paper deals with the entire solutions to a nonlocal dispersal bistable equation with spatio-temporal delay. Assuming that the equation has a traveling wave front with non-zero wave speed, we establish the existence of entire solutions with annihilating-fronts by using the comparison principle combined with explicit constructions of sub- and supersolutions. These entire solutions constitute a two-dimensional manifold and the traveling wave fronts belong to the boundary of the manifold. We also prove the uniqueness, Liapunov stability and continuous dependence on the shift parameters of the entire solutions.     

Localized solutions to the nonlinear wave equation for flexural motions of an elastic beam traveling in an air-filled tube

Steady-progressive-wave solutions are sought to the nonlinear wave equation derived previously [J. Fluids Struct. 16 (2002) 597] for flexural motions of an elastic beam traveling in an air-filled tube along its center axis at a subsonic speed. Fluid-structure interactions are taken into account through aerodynamic loading on the lateral surface of the beam subjected to small but finite deflection but end effects and viscous effects are neglected. Linear dispersion characteristics are first examined by exploiting the small ratio of the induced mass to the mass of the beam per unit length. Centered around the traveling speed of the beam, there exists such a narrow range of propagation velocity that the linear steady propagation is prohibited. In this range, it is revealed that some interesting nonlinear solutions exist. The periodic wavetrain is found to exist as the exact solution. Asymptotic analysis is then made by applying the method of multiple scales and the stationary nonlinear Schrödinger equation is derived for a complex amplitude. A monochromatic solution to this equation corresponds to the exact periodic solution. Imposing undisturbed boundary conditions at infinity, it is revealed that the localized solution exists as a result of balance between the linear instability and the nonlinearity. This solution is checked by solving the nonlinear equation numerically. It is further revealed that the amplitude-modulated wavetrain exists not only in the range of the velocity mentioned above but also outside of it.     

Bilinear Bäcklund transformation,Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Bäcklund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Bäcklund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.