Whitham Modulation Research Articles

Step-like initial value problem and Whitham modulation in fluid dynamics to a generalized derivative nonlinear Schrödinger equation

Step-like initial value problem and Whitham modulation in fluid dynamics to a generalized derivative nonlinear Schrödinger equation

Validity of Whitham's modulation equations for dissipative systems with a conservation law: Phase dynamics in a generalized Ginzburg-Landau system

It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms.

No-hair theorems for analogue black holes

We show that transonic one dimensional flows which are analogous to black holes obey no-hair theorems both at the level of linear perturbations and in non-linear regimes. Considering solutions of the Gross-Pitaevskii (or Korteweg-de Vries) equation, we show that stationary flows which are asymptotically uniform on both sides of the horizon are stable and act as attractors. Using Whitham's modulation theory, we analytically characterize the emitted waves when starting from uniform perturbations. Numerical simulations confirm the validity of this approximation and extend the results to more general perturbations and to the (non-integrable) cubic-quintic Gross-Pitaevskii equation. When considering time reversed flows that correspond to white holes, the asymptotically uniform flows are unstable to sufficiently large perturbations and emit either a macroscopic undulation in the supersonic side, or a non-linear superposition of soliton trains.

Whitham's modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky Equation

We study the spectral stability of periodic wave trains of the Korteweg-de Vries-Kuramoto-Sivashinsky equation which are, among many other applications, often used to describethe evo- lution of a thin liquid film flowing down an inclined ramp. More precisely, we show that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to side-band perturbations. Here, we use a direct Bloch ex- pansion method and spectral perturbation analysis instead of Evans function computations. We first establish, in our context, the now usual connection between first-order expansion of eigenvalues bi- furcating from the origin (both eigenvalue 0 and Floquet parameter 0) and the first-order Whitham's modulation system: the hyper- bolicity of such a system provides a necessary condition of spectral stability. Under a condition of strict hyperbolicity, we show that eigenvalues are indeed analytic in the neighborhood of the origin and that their expansion up to second order is connected to a vis- cous correction of the Whitham's equations. This, in turn, pro- vides new stability criteria. Finally, we study the Korteweg-de Vries limit: in this case, the domain of validity of the previous expansion shrinks to nothing and a new modulation theory is needed. The new modulation system consists of the Korteweg-de Vries modula- tion equations supplemented with a source term: relaxation limit in such a system provides, in turn, some stability criteria.

Dispersive dam-break and lock-exchange flows in a two-layer fluid

Dam-break and lock-exchange flows are considered in a Boussinesq two-layer fluid system in a uniform two-dimensional channel. The focus is on inviscid ‘weak’ dam breaks or lock exchanges, for which waves generated from the initial conditions do not break, but instead disperse in a so-called undular bore. The evolution of such flows can be described by the Miyata–Camassa–Choi (MCC) equations. Insight into solutions of the MCC equations is provided by the canonical form of their long wave limit, the two-layer shallow water equations, which can be related to their single-layer counterpart via a surjective map. The nature of this surjective map illustrates that whilst some Riemann-type initial-value problems (dam breaks) are analogous to those in the single-layer problem, others (lock exchanges) are not. Previous descriptions of MCC waves of permanent form (cnoidal and solitary waves) are generalised, including a description of the effects of a regularising surface tension. The wave solutions allow the application of a technique due to El's approach, based on Whitham's modulation theory, which is used to determine key features of the expanding undular bore as a function of the initial conditions. A typical dam-break flow consists of a leftwards-propagating simple rarefaction wave and a rightward-propagating simple undular bore. The leading and trailing edge speeds, leading edge solitary wave amplitude and trailing edge linear wavelength are determined for the undular bore. Lock-exchange flows, for which the initial interface shape crosses the mid-depth of the channel, by contrast, are found to be more complex, and depending on the value of the surface tension parameter may include ‘solibores’ or fronts connecting two distinct regimes of long-wave behaviour. All of the results presented are informed and verified by numerical solutions of the MCC equations.

Micro–macro transition in the atomic chain via Whitham's modulation equation

The subject matter of this paper is the thermodynamic description of the nonlinear atomic chain with temperature. For this reason we consider special approximate solutions of Newton's equations, in which the atoms perform microscopic oscillations in the form of modulated travelling waves. We start with an existence result for periodic travelling waves with arbitrary large amplitudes and study several examples including the harmonic chain, the hard sphere model and the small-amplitude approximation. Then we discuss the thermodynamic properties of travelling waves and derive the corresponding Gibbs equation. Afterwards we focus on the macroscopic evolution of modulated travelling waves. For this purpose we apply Whitham's modulation theory to the atomic chain and derive the modulation equation, which turns out to be a system of four macroscopic conservation laws. The last part is devoted to the justification problem: we state a conjecture for the general case and prove this conjecture for the harmonic chain and the hard sphere model.

Asymptotic solutions of the nonlocal nonlinear Schrödinger equation in the limit of small dispersion

A method is developed for solving the initial value problem of the nonlocal nonlinear Schrödinger (NLS) equation in the limit of small dispersion. Whitham's modulation theory is used to characterize the main feature of the solution in terms of the single-phase periodic solution of the nonlocal NLS equation with the slowly varying wave parameters. The modulation equations for these parameters are derived by averaging the local conservation laws. A novel feature of the modulation equations is that they can be decoupled into the form of the integrable Hopf equation. An explicit example of the solution is exhibited for a step initial condition.

The modulation of nonlinear periodic wavetrains by dissipative terms in the Korteweg-de Vries equation

A variable coefficient, perturbed Korteweg-de Vries equation is considered, whose coefficients are slowly-varying in time, and for which the perturbation term represents various forms of dissipation. We construct asymptotic solutions for modulated periodic wavetrains using a multiple-scale expansion. The modulations are described by a nonlinear, non-homogeneous system of first-order partial differential equations for the parameters of the wavetrain that describes an extension to Whitham's modulation theory. We deal specifically with three forms of frictional terms representing (a) linear damping, (b) KdV-Burgers damping, and (c) boundary-layer damping. The initial condition is a step discontinuity which evolves into a disturbance resembling an undular bore. Then, in particular, we use this modulation theory to study the behaviour at and near the front and rear of the bore.

On the modulation of solutions to scalar lax equations according to Whitham's procedure

Flaschka, Forest and McLaughlin have demonstrated that Whitham's modulation equations for generic “ g-gap” solutions of the Korteweg-de Vries equation can be expressed as the variation of certain meromorphic differentials over a family of Riemann surfaces. Here we derive the analogous result for an arbitrary scalar Lax equation.

Derivation of the nonlinear Schr�dinger Equation for shoaling wave-groups

A nonlinear Schrodinger equation with varying coefficients, describing the evolution of surface wave groups moving over an uneven bottom, is derived from Whitham's modulation equations. The derivation yields new expressions for the wave-induced mean flow field.