Validity of Whitham’s Equations for the Modulation of Periodic Traveling Waves in the NLS Equation

We prove that slow modulations in time and space of periodic wave trains of the NLS equation can be approximated via solutions of Whitham’s equations associated with the wave train. The error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like existence and uniqueness theorem, and energy estimates.     

Existence and multiplicity of wave trains in 2D lattices

We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov–Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and p:q$p:q$ resonance, respectively. We show the impact of the direction θ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of traveling waves of the original two-dimensional lattice in the direction θ of propagation satisfying tan θ is rational.     

Pressure transients generated when high-speed trains pass in a tunnel

An analysis is made of the pressure transients generated when two high-speed trains meet in a tunnel. Safe operation at speeds exceeding about 300 km/h usually demands that the hydraulicblockage produced by a train should be small. The problem canthen be formulated in terms of the scattering of the potentialflow near field of each train by the moving surface of theother train, and this permits the derivation of closed formrepresentations of the unsteady pressure in the special caseof ‘snub nosed’ trains in a tunnel of semicircularcross-section. This solution is used to devise a general procedurefor calculating pressure transients generated by trains ofarbitrary nose profiles in tunnels of arbitrary cross-sectionalshape in terms of a knowledge of the local incompressible potentialflow produced by each train travelling separately in the tunnel.Numerical results indicate that at train Mach numbers exceeding0.25the amplitudes of the pressure transients generated bymeeting trains will typically exceed about 25 per cent of theamplitude of the compression wave produced when a train entersor leaves the tunnel. Received 12 April, 1999. Revised 13 March, 2000.     

Some Explicit Resonating Waves in Weakly Nonlinear Gas Dynamics

Equations governing leading order wave amplitudes of resonating almost periodic wave trains in weakly nonlinear acoustics have been obtained by Majda and Rosales [Stud. Appl. Math. 71:149–179 (1984)]. These equations consist of a pair of Burgers equations coupled through an integral term with a known kernel. Numerical experiments reported by Majda, Rosales, and Schonbek have suggested the existence of smooth solutions of this system whose components consist of traveling waves moving in opposite directions. For the simplest cosine kernel, explicit formulae are given here for such resonating wave solutions. There is a wave of maximum amplitude with a “peak.” For more general kernels, small amplitude resonating waves are constructed via bifurcation.     

Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation

Bethuel et al.  and  and Chiron and Rousset [3] gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the NLS equation can approximately be described via solutions of the KdV equation associated with the wave train. Here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of ,  and . Our error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like method, and energy estimates.     

Triggered Fronts in the Complex Ginzburg Landau Equation

We study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex Ginzburg–Landau equation. We model the trigger by a spatial inhomogeneity moving with constant speed. In the comoving frame, the trivial state is unstable to the left of the trigger and stable to the right. At the trigger location, spatio-temporally periodic wave trains nucleate. Our results show existence of coherent, “heteroclinic” profiles when the speed of the trigger is slightly below the speed of a free front in the unstable medium. Our results also give expansions for the wavenumber of wave trains selected by these coherent fronts. A numerical comparison yields very good agreement with observations, even for moderate trigger speeds. Technically, our results provide a heteroclinic bifurcation study involving an equilibrium with an algebraically double pair of complex eigenvalues. We use geometric desingularization and invariant foliations to describe the unfolding. Leading-order terms are determined by a condition of oscillations in a projectivized flow, which can be found by intersecting absolute spectra with the imaginary axis.     

Solitary and Periodic Solutions of Nonlinear Nonintegrable Equations

The singular manifold method and partial fraction decomposition allow one to find some special solutions of nonintegrable partial differential equations (PDE) in the form of solitary waves, traveling wave fronts, and periodic pulse trains. The truncated Painlevé expansion is used to reduce a nonlinear PDE to a multilinear form. Some special solutions of the latter equation represent solitary waves and traveling wave fronts of the original PDE. The partial fraction decomposition is used to obtain a periodic wave train solution as an infinite superposition of the "corrected" solitary waves.     

A note on the existence of periodic travelling wave solutions with large periods in generalized reaction-diffusion systems

We show that the existence of wave trains with high velocity of generalized reaction-diffusion equations can be easily established by using a theorem of D. V. Anosov on the existence of periodic solutions of singularly perturbed differential systems.     

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.     

A Canonical System of lntegrodifferential Equations Arising in Resonant Nonlinear Acoustics

In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order. In earlier work the first two authors and J. Hunter developed simplified asymptotic equations describing this resonant interaction. In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy. Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics. These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations. Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system.     

Pulse propagation in a viscoelastic tube containing a viscous liquid

Previous attempts at analysing tube propagation in a viscoelastic tube containing a viscous liquid have concentrated on examining sinusoidal wave trains of infinite length. This paper takes viscosity into account in the formulation of initial and boundary value problems which mimic experimental configurations.     

Evans functions for periodic waves on infinite cylindrical domains

Using Galerkin approximations, an Evans function for spatially periodic waves on infinite cylindrical domains is constructed. It is also shown that the Evans function can be used to define a parity index for periodic waves that detects whether the wave admits an odd number of real unstable eigenvalues. This parity index depends only on local information for the existence problem of the wave: in particular, it uses information about the linear dispersion relation near zero and the orientability of the unstable and stable manifolds along the nonlinear wave. The results are applied to small-amplitude wave trains for a scalar equation on an infinite strip.     

TARGET PATTERNS AND SPIRAL WAVES IN REACTION-DIFFUSION SYSTEMS WITH LIMIT CYCLE KINETICSAND WEAK DIFFUSION

1.IntroductionTargetpatternsandspiralwavesarecommonlyobservedincertainmodelsofchemicalandbiologicalsystemssuchastheBelousov-ZhabotinskiireactionandthesocialamoebasDictyosteliumdiscoideium(of.11--4]andthereferencestherein).Thesesystemsaregovernedbyachemicalorbiologicalreactionandspatialdiffusion,i.e.reaction-diffusionequations.Generallyspeaking,atargetpatternisasetofconcentricringsofconstantconcentrationeachmovingoutward.Alonganyradialline,thepatternbehavesasymptoticallylikeaperiodictravellin…     

Transverse Orbital Stability of Periodic Traveling Waves for Nonlinear Klein–Gordon Equations

In this paper, we establish the orbital stability of a class of spatially periodic wave train solutions to multidimensional nonlinear Klein–Gordon equations with periodic potential. We show that the orbit generated by the one‐dimensional wave train is stable under the flow of the multidimensional equation under perturbations which are, on one hand, coperiodic with respect to the translation or Galilean variable of propagation, and, on the other hand, periodic (but not necessarily coperiodic) with respect to the transverse directions. That is, we show their transverse orbital stability. The class of periodic wave trains under consideration is the family of subluminal rotational waves, which are periodic in the momentum but unbounded in their position.     

A Note on Resonance and Nonlinear Dispersive Waves

The general theory for the slow dispersion of nonlinear wave trains, first studied by Whitham, is applied to a wave train, which, in the weakly nonlinear limit, exhibits resonant singularities. Numerical and perturbation methods are used to develop singly periodic solutions both away from and near all such critical values. Similarly, the equations governing the slow modulations of such a system are found by asymptotic analysis. The expansions are found to be valid so long as the wave train is sufficiently nonlinear. These ideas should be applicable to other problems where resonant singularities arise, in particular, multiphase modes.     

Nonlinear Amplitude Evolution of Baroclinic Wave Trains and Wave Packets

The weakly nonlinear theory of baroclinic wave trains and wave packets is examined by the use of systematic expansion procedures in appropriate powers of a small parameter measuring the supercriticality according to linear theory; well-known multiple scaling techniques are employed. Crucial importance is ascribed to the magnitude of parameters measuring dissipation and dispersion relative to each other and to the supercriticality, and equations describing the slow evolution in space and time of the wave amplitude are established for a range of parameter values. For vanishingly small dissipation the wave train equations have straightforward oscillatory solutions, dependent on initial conditions, and for large dissipation steady equilibration, independent of initial conditions, is predicted. For moderately small dissipation, however, a wide variety of behaviors is possible—including steady equilibration, single and multiple periodicity, and aperiodicity—in the solutions of the equations, which are recognizable as generalisations of the well-known Lorenz attractor equations. Equations describing the evolution of wave packets take a variety of forms; for vanishingly small dissipation or for large dissipation, they are essentially parabolic and of nonlinear Schrödinger type, whilst for moderate dissipation they are of Lorenz type, modified by spatial variations. Solutions of a number of these equations are discussed and compared, where appropriate, with experimental results.     

Nonlinear Rayleigh-Taylor instability in magnetic fluids with uniform horizontal and vertical magnetic fields

A weakly nonlinear evolution of two dimensional wave packets on the surface of a magnetic fluid in the presence of an uniform magnetic field is presented, taking into account the surface tension. The method used is that of multiple scales to derive two partial differential equations. These differential equations can be combined to yield two alternate nonlinear Schrödinger equations. The first equation is valid near the cutoff wavenumber while the second equation is used to show that stability of uniform wave trains depends on the wavenumber, the density, the surface tension and the magnetic field. At the critical point, a generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. From the latter equation, the various stability crteria are obtained.     

Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

Inverse spectral theory is used to prescribe and study equations for the slow modulations of N-phase wave trains for the Korteweg-de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single-phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first-order terms.     

Oscillatory reaction-diffusion equations with temporally varying parameters

Periodic wave trains are the generic one-dimensional solution form for reaction-diffusion equations with a limit cycle in the kinetics. Such systems are widely used as models for oscillatory phenomena in chemistry, ecology, and cell biology. In this paper, we study the way in which periodic wave solutions of such systems are modified by periodic forcing of kinetic parameters. Such forcing will occur in many ecological applications due to seasonal variations. We study temporal forcing in detail for systems of two reaction diffusion equations close to a supercritical Hopf bifurcation in the kinetics, with equal diffusion coefficients. In this case, the kinetics can be approximated by the Hopf normal form, giving reaction-diffusion equations of λ-ω type. Numerical simulations show that a temporal variation in the kinetic parameters causes the wave train amplitude to oscillate in time, whereas in the absence of any temporal forcing, this wave train amplitude is constant. Exploiting the mathematical simplicity of the λ-ω form, we derive analytically an approximation to the amplitude of the wave train oscillations with small forcing. We show that the amplitude of these oscillations depends crucially on the period of forcing.