30 research outputs found
Flat bidifferential ideals and semihamiltonian 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.Comment: 18 pages. v5: formula (36) corrected; minor change
Wave envelopes with second-order spatiotemporal dispersion : I. Bright Kerr solitons and cnoidal waves
We propose a simple scalar model for describing pulse phenomena beyond the conventional slowly-varying envelope approximation. The generic governing equation has a cubic nonlinearity and we focus here mainly on contexts involving anomalous group-velocity dispersion. Pulse propagation turns out to be a problem firmly rooted in frames-of-reference considerations. The transformation properties of the new model and its space-time structure are explored in detail. Two distinct representations of exact analytical solitons and their associated conservation laws (in both integral and algebraic forms) are presented, and a range of new predictions is made. We also report cnoidal waves of the governing nonlinear equation. Crucially, conventional pulse theory is shown to emerge as a limit of the more general formulation. Extensive simulations examine the role of the new solitons as robust attractors
Wave envelopes with second-order spatiotemporal dispersion: II. Modulational instabilities and dark Kerr solitons
A simple scalar model for describing spatiotemporal dispersion of pulses, beyond the classic âslowly-varying envelopes + Galilean boostâ approach, is studied. The governing equation has a cubic nonlinearity and we focus here mainly on contexts with normal group-velocity dispersion. A complete analysis of continuous waves is reported, including their dispersion relations and modulational instability characteristics. We also present a detailed derivation of exact analytical dark solitons, obtained by combining direct-integration methods with geometrical transformations. Classic results from conventional pulse theory are recovered as-ymptotically from the spatiotemporal formulation. Numerical simulations test new theoretical predictions for modulational instability, and examine the robustness of spatiotemporal dark solitons against perturbations to their local pulse shape
Singular sectors of the 1-layer Benney and dToda systems and their interrelations
Complete description of the singular sectors of the 1-layer Benney system
(classical long wave equation) and dToda system is presented. Associated
Euler-Poisson-Darboux equations E(1/2,1/2) and E(-1/2,-1/2) are the main tool
in the analysis. A complete list of solutions of the 1-layer Benney system
depending on two parameters and belonging to the singular sector is given.
Relation between Euler-Poisson-Darboux equations E(a,a) with opposite sign of a
is discussed.Comment: 15 pages; Theor. Mathem. Physics, 201
Stable one-dimensional periodic waves in Kerr-type saturable and quadratic nonlinear media
We review the latest progress and properties of the families of bright and
dark one-dimensional periodic waves propagating in saturable Kerr-type and
quadratic nonlinear media. We show how saturation of the nonlinear response
results in appearance of stability (instability) bands in focusing (defocusing)
medium, which is in sharp contrast with the properties of periodic waves in
Kerr media. One of the key results discovered is the stabilization of
multicolor periodic waves in quadratic media. In particular, dark-type waves
are shown to be metastable, while bright-type waves are completely stable in a
broad range of energy flows and material parameters. This yields the first
known example of completely stable periodic wave patterns propagating in
conservative uniform media supporting bright solitons. Such results open the
way to the experimental observation of the corresponding self-sustained
periodic wave patterns.Comment: 29 pages, 10 figure
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
Stable multicolor periodic-wave arrays
We study the existence and stability of cnoidal periodic wave arrays
propagating in uniform quadratic nonlinear media and discover that they become
completely stable above a threshold light intensity. To the best of our
knowledge, this is the first example in physics of completely stable periodic
wave patterns propagating in conservative uniform media supporting bright
solitons.Comment: 12 pages, 3 figure
On critical behaviour in systems of Hamiltonian partial differential equations
We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture