On Whitham theory for perturbed integrable equations

Whitham theory of modulations is developed for periodic waves described by nonlinear wave equations integrable by the inverse scattering transform method associated with $2\times2$ matrix or second order scalar spectral problems. The theory is illustrated by derivation of the Whitham equations for perturbed Korteweg-de Vries equation and nonlinear Schr\"odinger equation with linear damping.Comment: 17 pages, no figure

Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory

We derive an asymptotic formula for the amplitude distribution in a fully nonlinear shallow-water solitary wave train which is formed as the long-time outcome of the initial-value problem for the Su-Gardner (or one-dimensional Green-Naghdi) system. Our analysis is based on the properties of the characteristics of the associated Whitham modulation system which describes an intermediate "undular bore" stage of the evolution. The resulting formula represents a "non-integrable" analogue of the well-known semi-classical distribution for the Korteweg-de Vries equation, which is usually obtained through the inverse scattering transform. Our analytical results are shown to agree with the results of direct numerical simulations of the Su-Gardner system. Our analysis can be generalised to other weakly dispersive, fully nonlinear systems which are not necessarily completely integrable.Comment: 25 pages, 7 figure

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 defocusing nonlinear Schr\"odinger (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.Comment: 45 pages, 23 figures, minor revisio

Analytic model for a frictional shallow-water undular bore

We use the integrable Kaup-Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by friction. This is derived in Riemann variables using a modified finite-gap integration technique for the AKNS scheme. The Whitham system is then reduced to a simple first-order differential equation which is integrated numerically to obtain an asymptotic profile of the undular bore, with the local oscillatory structure described by the periodic solution of the unperturbed Kaup-Boussinesq system. This solution of the Whitham equations is shown to be consistent with certain jump conditions following directly from conservation laws for the original system. A comparison is made with the recently studied dissipationless case for the same system, where the undular bore is unsteady.Comment: 24 page

Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type

We study the Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion and with monotonically increasing initial data using the Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero dispersion limit, is obtained using the steepest descent method for oscillatory Riemann-Hilbert problems. The asymptotic solution is completely described by a scalar function \g that satisfies a scalar RH problem and a set of algebraic equations constrained by algebraic inequalities. The scalar function \g is equivalent to the solution of the Lax-Levermore maximization problem. The solution of the set of algebraic equations satisfies the Whitham equations. We show that the scalar function \g and the Lax-Levermore maximizer can be expressed as the solution of a linear overdetermined system of equations of Euler-Poisson-Darboux type. We also show that the set of algebraic equations and algebraic inequalities can be expressed in terms of the solution of a different set of linear overdetermined systems of equations of Euler-Poisson-Darboux type. Furthermore we show that the set of algebraic equations is equivalent to the classical solution of the Whitham equations expressed by the hodograph transformation.Comment: 32 pages, 1 figure, latex2

The thermodynamic limit of the Whitham equations

The infinite-genus limit of the KdV-Whitham equations is derived. The limit involves special scaling for the associated spectral surface such that the integrated density of states remains finite as $N \to \infty$ (the thermodynamic type limit). The limiting integro-differential system describes slow evolution of the density of states and can be regarded as the kinetic equation for a soliton gas

Solitons of the Resonant Nonlinear Schrodinger Equation with Nontrivial Boundary Conditions and Hirota Bilinear Method

Physically relevant soliton solutions of the resonant nonlinear Schrodinger (RNLS) equation with nontrivial boundary conditions, recently proposed for description of uniaxial waves in a cold collisionless plasma, are considered in the Hirota bilinear approach. By the Madelung representation, the model is transformed to the reaction-diffusion analog of the NLS equation for which the bilinear representation, soliton solutions and their mutual interactions are studied.Comment: 15 pages, 1 figure, talk presented in Workshop `Nonlinear Physics IV: Theory and Experiment`, 22-30 June 2006, Gallipoli, Ital

Unsteady undular bores in fully nonlinear shallow-water theory

We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for ``depth ratios'' across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9 figure

The theory of optical dispersive shock waves in photorefractive media

The theory of optical dispersive shocks generated in propagation of light beams through photorefractive media is developed. Full one-dimensional analytical theory based on the Whitham modulation approach is given for the simplest case of sharp step-like initial discontinuity in a beam with one-dimensional strip-like geometry. This approach is confirmed by numerical simulations which are extended also to beams with cylindrical symmetry. The theory explains recent experiments where such dispersive shock waves have been observed.Comment: 26 page

Resolution of a shock in hyperbolic systems modified by weak dispersion

We present a way to deal with dispersion-dominated ``shock-type'' transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic regularized by a small amount of dispersion. The analysis is performed by assuming that, the dispersive shock transition between two different constant states can be modelled by an expansion fan solution of the associated modulation (Whitham) system for the short-wavelength nonlinear oscillations in the transition region (the so-called Gurevich -- Pitaevskii problem). We consider as single-wave so bi-directional systems. The main mathematical assumption is that of hyperbolicity of the Whitham system for the solutions of our interest. By using general properties of the Whitham averaging for a certain class of nonlinear dispersive systems and specific features of the Cauchy data prescription on characteristics we derive a set of transition conditions for the dispersive shock, actually bypassing full integration of the modulation equations. Along with model KdV and mKdV examples, we consider a non-integrable system describing fully nonlinear ion-acoustic waves in collisionless plasma. In all cases our transition conditions are in complete agreement with previous analytical and numerical results.Comment: 56 pages, 5 figures. Misprints corrected. References adde