Whitham Modulation Equations Research Articles

Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate.

We provide a classification of the possible flows of two-component Bose-Einstein condensates evolving from initially discontinuous profiles. We consider the situation where the dynamics can be reduced to the consideration of a single polarization mode (also denoted as "magnetic excitation") obeying a system of equations equivalent to the Landau-Lifshitz equation for an easy-plane ferromagnet. We present the full set of one-phase periodic solutions. The corresponding Whitham modulation equations are obtained together with formulas connecting their solutions with the Riemann invariants of the modulation equations. The problem is not genuinely nonlinear, and this results in a non-single-valued mapping of the solutions of the Whitham equations with physical wave patterns as well as the appearance of interesting elements-contact dispersive shock waves-that are absent in more standard, genuinely nonlinear situations. Our analytic results are confirmed by numerical simulations.

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion.

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks, which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.

Whitham modulation theory for the Ostrovsky equation.

This paper derives the Whitham modulation equations for the Ostrovsky equation. The equations are then used to analyse localized cnoidal wavepacket solutions of the Ostrovsky equation in the weak rotation limit. The analysis is split into two main parameter regimes: the Ostrovsky equation with normal dispersion relevant to typical oceanic parameters and the Ostrovsky equation with anomalous dispersion relevant to strongly sheared oceanic flows and other physical systems. For anomalous dispersion a new steady, symmetric cnoidal wavepacket solution is presented. The new wavepacket can be represented as a solution of the modulation equations and an analytical solution for the outer solution of the wavepacket is given. For normal dispersion the modulation equations are used to describe the unsteady finite-amplitude wavepacket solutions produced from the rotation-induced decay of a Korteweg-de Vries solitary wave. Again, an analytical solution for the outer solution can be given. The centre of the wavepacket closely approximates a train of solitary waves with the results suggesting that the unsteady wavepacket is a localized, modulated cnoidal wavetrain. The formation of wavepackets from solitary wave initial conditions is considered, contrasting the rapid formation of the packets in anomalous dispersion with the slower formation of unsteady packets under normal dispersion.

On the Elliptic-Hyperbolic Transition in Whitham Modulation Theory

The dispersionless Whitham modulation equations in one space dimension and time are generically hyperbolic or elliptic, and breakdown at the transition, which is a curve in the frequency-wavenumber plane. In this paper, the modulation theory is reformulated with a slow phase and different scalings resulting in a phase modulation equation near the singular curves which is a geometric form of the two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multiperiodic, quasiperiodic and multi-pulse localized solutions. This theory shows that the elliptic-hyperbolic transition is a rich source of complex behaviour in nonlinear wave fields. There are several examples of these transition curves in the literature to which the theory applies. For illustration the theory is applied to the complex nonlinear Klein-Gordon equation which has two singular curves in the manifold of periodic travelling waves.

Modulations of viscous fluid conduit periodic waves

Modulated periodic interfacial waves along a conduit of viscous liquid are explored using nonlinear wave modulation theory and numerical methods. Large-amplitude periodic-wave modulation (Whitham) theory does not require integrability of the underlying model equation, yet often either integrable equations are studied or the full extent of Whitham theory is not developed. Periodic wave solutions of the nonlinear, dispersive, non-integrable conduit equation are characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the nonlinear Schrödinger (NLS) equation are derived, depending on the carrier wavenumber. Dark and bright envelope solitons are found to persist in long-time numerical solutions of the conduit equation, providing numerical evidence for the existence of strongly nonlinear, large-amplitude envelope solitons. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted and observed. In the large-amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. The viscous fluid conduit system is a mathematically tractable, experimentally viable model system for wide-ranging nonlinear, dispersive wave dynamics.

Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations

Dispersive shock waves (DSWs) in the Kadomtsev–Petviashvili (KP) equation and two dimensional Benjamin–Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time (2+1) dimensions to finding DSW solutions of (1+1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg–de Vries (cKdV) and cylindrical Benjamin–Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the (2+1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1+1) dimensional equations.

The propagation of internal undular bores over variable topography

In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and can be modelled by equations of the Korteweg–de Vries type. Typically they occur in regions of variable bottom topography when the variable-coefficient Korteweg–de Vries equation is an appropriate model. Of special interest is the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here we examine the same situation for an undular bore, represented by a modulated periodic wave train. Numerical simulations and some asymptotic analysis based on Whitham modulation equations show that the leading solitary waves in the undular bore are destroyed and replaced by a developing rarefaction wave supporting emerging solitary waves of the opposite polarity. In contrast the rear of the undular bore emerges with the same shape, but with reduced wave amplitudes, a shorter overall length scale and moves more slowly.

Whitham modulation equations, coalescing characteristics, and dispersive Boussinesq dynamics

Whitham modulation theory with degeneracy in wave action is considered. The case where all components of the wave action conservation law, when evaluated on a family of periodic travelling waves, have vanishing derivative with respect to wavenumber is considered. It is shown that Whitham modulation equations morph, on a slower time scale, into the two way Boussinesq equation. Both the 1+1 and 2+1 cases are considered. The resulting Boussinesq equation arises in a universal form, in that the coefficients are determined from the abstract properties of the Lagrangian and do not depend on particular equations. One curious by-product of the analysis is that the theory can be used to confirm that the two-way Boussinesq equation is not a valid model in shallow water hydrodynamics. Modulation of nonlinear travelling waves of the complex Klein–Gordon equation is used to illustrate the theory.

Whitham theory for perturbed Korteweg–de Vries equation

Original Whitham’s method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg–de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of right-hand sides in the modulation equations so that they become non-uniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained in both cases. The essential difference between them is illustrated by an example of so-called ‘generalized Korteweg–de Vries equation’. Method of finding steady-state solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.

Breakdown of the Whitham Modulation Theory and the Emergence of Dispersion

The Whitham modulation theory for periodic traveling waves of PDEs generated by a Lagrangian produces first‐order dispersionless PDEs that are, generically, either hyperbolic or elliptic. In this paper, degeneracy of the Whitham equations is considered where one of the characteristic speeds is zero. In this case, the Whitham equations are no longer valid. Reformulation and rescaling show that conservation of wave action morphs into the Korteweg–de Vries (KdV) equation on a longer time scale thereby generating dispersion in the Whitham modulation equations even for finite amplitude waves.

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation

In this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized Kuramoto–Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch–She–Thual, Bar–Nepomnyashchy, Chang–Demekhin–Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader’s convenience, we include in an appendix the corresponding treatment of the Swift–Hohenberg equation, a nonconservative counterpart of the generalized Kuramoto–Sivashinsky equation for which the nonlinear stability analysis is considerably simpler, together with numerical Evans function analyses extending spectral stability analyses of Mielke and Schneider.

Dispersive shock waves in viscously deformable media

AbstractThe viscously dominated, low-Reynolds-number dynamics of multi-phase, compacting media can lead to nonlinear, dissipationless/dispersive behaviour when viewed appropriately. In these systems, nonlinear self-steepening competes with wave dispersion, giving rise to dispersive shock waves (DSWs). Example systems considered here include magma migration through the mantle as well as the buoyant ascent of a low-density fluid through a viscously deformable conduit. These flows are modelled by a third-order, degenerate, dispersive, nonlinear wave equation for the porosity (magma volume fraction) or cross-sectional area, respectively. Whitham averaging theory for step initial conditions is used to compute analytical, closed-form predictions for the DSW speeds and the leading edge amplitude in terms of the constitutive parameters and initial jump height. Novel physical behaviours are identified including backflow and DSW implosion for initial jumps sufficient to cause gradient catastrophe in the Whitham modulation equations. Theoretical predictions are shown to be in excellent agreement with long-time numerical simulations for the case of small- to moderate-amplitude DSWs. Verifiable criteria identifying the breakdown of this modulation theory in the large jump regime, applicable to a wide class of DSW problems, are presented.

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.

Refraction of dispersive shock waves

We study a dispersive counterpart of the classical gas dynamics problem of the interaction of a shock wave with a counter-propagating simple rarefaction wave, often referred to as the shock wave refraction. The refraction of a one-dimensional dispersive shock wave (DSW) due to its head-on collision with the centred rarefaction wave (RW) is considered in the framework of the defocusing nonlinear Schrödinger (NLS) equation. For the integrable cubic nonlinearity case we present a full asymptotic description of the DSW refraction by constructing appropriate exact solutions of the Whitham modulation equations in Riemann invariants. For the NLS equation with saturable nonlinearity, whose modulation system does not possess Riemann invariants, we take advantage of the recently developed method for the DSW description in non-integrable dispersive systems to obtain main physical parameters of the DSW refraction. The key features of the DSW-RW interaction predicted by our modulation theory analysis are confirmed by direct numerical solutions of the full dispersive problem.

Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg-de Vries Equation

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long-wavelength perturbations. Specifically, we extend the work of Bronski and Johnson by demonstrating that the homogenized system describing the mean behavior of a slow modulation (WKB) approximation of the solution correctly describes the linearized dispersion relation near zero frequency of the linearized equations about the background periodic wave. The latter has been shown by rigorous Evans function techniques to control the spectral stability near the origin, that is, stability to slow modulations of the underlying solution. In particular, through our derivation of the WKB approximation we generalize the modulation expansion of Whitham for the KdV to a more general class of equations which admit periodic waves with nonzero mean. As a consequence, we will show that, assuming a particular nondegeneracy condition, spectral stability near the origin is equivalent with the local well-posedness of the Whitham system.

Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle

Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional (2D) defocusing nonlinear Schrödinger (NLS) equation. This problem is of fundamental importance as a dispersive analog of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed is sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear "ship-wave" pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.

Transcritical flow of a Bose-Einstein condensate through a penetrable barrier

The problem of the transcritical flow of a Bose-Einstein condensate through a wide repulsive penetrable barrier is studied analytically using the combination of the localized "hydraulic" solution of the 1D Gross-Pitaevskii equation and the solutions of the Whitham modulation equations describing the resolution of the upstream and downstream discontinuities through dispersive shocks. It is shown that within the physically reasonable range of parameters the downstream dispersive shock is attached to the barrier and effectively represents the train of very slow dark solitons, which can be observed in experiments. The rate of the soliton emission, the amplitudes of the solitons in the train and the drag force are determined in terms of the BEC oncoming flow velocity and the strength of the potential barrier. A good agreement with direct numerical solutions is demonstrated. Connection with recent experiments is discussed.

Interactions of dispersive shock waves

Collisions and interactions of dispersive shock waves in defocusing (repulsive) nonlinear Schrödinger type systems are investigated analytically and numerically. Two canonical cases are considered. In one case, two counterpropagating dispersive shock waves experience a head-on collision, interact and eventually exit the interaction region with larger amplitudes and altered speeds. In the other case, a fast dispersive shock overtakes a slower one, giving rise to an interaction. Eventually the two merge into a single dispersive shock wave. In both cases, the interaction region is described by a modulated, quasi-periodic two-phase solution of the nonlinear Schrödinger equation. The boundaries between the background density, dispersive shock waves and their interaction region are calculated by solving the Whitham modulation equations. These asymptotic results are in excellent agreement with full numerical simulations. It is further shown that the interactions of two dispersive shock waves have some qualitative similarities to the interactions of two classical shock waves.

Whitham method for the Benjamin-Ono-Burgers equation and dispersive shocks

The Whitham modulation equations for the parameters of a periodic solution are derived using the generalized Lagrangian approach for the case of the damped Benjamin-Ono equation. The structure of the dispersive shock is considered in this method.

Flat bidifferential ideals and semi-Hamiltonian PDEs

In this paper we consider a class of semihamiltonian systems characterized by the existence of a special conservation law. The density and the current of this conservation law satisfy a second order system of PDEs which has a natural interpretation in the theory of flat bifferential ideals. The class of systems we consider contains important well-known examples of semihamiltonian systems. Other examples, like genus 1 Whitham modulation equations for KdV, are related to this class by a reciprocal trasformation.