Higher order terms in multiscale expansions: a linearized KdV hierarchy

We consider a wide class of model equations, able to describe wave propagation in dispersive nonlinear media. The Korteweg-de Vries (KdV) equation is derived in this general frame under some conditions, the physical meanings of which are clarified. It is obtained as usual at leading order in some multiscale expansion. The higher order terms in this expansion are studied making use of a multi-time formalism and imposing the condition that the main term satisfies the whole KdV hierarchy. The evolution of the higher order terms with respect to the higher order time variables can be described through the introduction of a linearized KdV hierarchy. This allows one to give an expression of the higher order time derivatives that appear in the right hand member of the perturbative expansion equations, to show that overall the higher order terms do not produce any secularity and to prove that the formal expansion contains only bounded terms.Comment: arxiv version is already officia

Mixed perturbative expansion: the validity of a model for the cascading

A new type of perturbative expansion is built in order to give a rigorous derivation and to clarify the range of validity of some commonly used model equations. This model describes the evolution of the modulation of two short and localized pulses, fundamental and second harmonic, propagating together in a bulk uniaxial crystal with non-vanishing second order susceptibility $\chi^(2)$ and interacting through the nonlinear effect known as ``cascading'' in nonlinear optics. The perturbative method mixes a multi-scale expansion with a power series expansion of the susceptibility, and must be carefully adapted to the physical situation. It allows the determination of the physical conditions under which the model is valid: the order of magnitude of the walk-off, phase-mismatch,and anisotropy must have determined values.Comment: arxiv version is already officia

The KdV hierarchy and the propagation of solitons on very long distances

The Korteweg-de Vries (KdV) equation is first derived from a general system of partial differential equations. An analysis of the linearized KdV equation satisfied by the higher order amplitudes shows that the secular-producing terms in this equation are the derivatives of the conserved densities of KdV. Using the multi-time formalism, we prove that the propagation on very long distances is governed by all equations of the KdV hierarchy. We compute the soliton solution of the complete hierarchy, which allows to give a criterion for the existence of the soliton

Solitons in ferromagnets and the KdV Hierarchy

The higher order terms in the perturbative expansion that describes KdV solitons propagation in ferromagnetic materials are considered. They satisfy inhomogeneous linearized KdV equations, explicitly written down. The parity and homogeneity properties of the expansion show that half of these equations admit a zero solution. Long time propagation is investigated, through the consideration of the unbounded or secular solutions and a multi-time expansion. It is governed by all equations of the KdV Hierarchy. Major result of the paper is that the multi-scale expansion can be achieved up to any order with all its terms bounded, what is a necessary condition for its convergence

Interaction of two solitary waves in a ferromagnet

A type of solitary wave in a ferromagnet is found by a multiscale expansion method; it obeys the completely integrable Korteweg-de Vries equation. The interaction between a wave of this propagation mode and another known mode that also allows soliton propagation is studied. The equations describing the interaction are derived using a multiscale expansion method and then reduced to an integral form, and solved explicitly for particular initial data for which one of the waves can be considered as a soliton. A phase shift of this soliton appears. Transmission and reflexion coefficients are computed for the second wave

Bidimensional optical solitons in a quadratic medium

The modulation evolution of a short localized optical pulse in a crystal belonging to one of the classes 42m, 43m, 3m, 6mm, and with a non-vanishing second-order nonlinearity, is considered. In (2 + 1) dimensions, the partial differential system accounting for it can be reduced to the completely integrable Davey–Stewartson system, if some conditions are satisfied. The first integrability condition represents a balance between the third-order Kerr effect and the cascaded second-order nonlinearities, while the second condition is an equilibrium between the dispersion and the kinetic factor of the electro-optic–optical rectification wave interaction. For anomalous dispersion, the obtained Davey–Stewartson system is of the type I, that admits localized soliton solutions. Lump solution, algebraically decaying in all directions, exist in any case satisfying the above conditions

Electromagnetic waves in ferromagnets: a Davey-Stewartson-type model

We examine the nonlinear modulation of an electromagnetic localized pulse in a saturated bulk ferromagnetic medium. It is seen that the evolution of the pulse shape is governed by a three-dimensional generalization of the Davey–Stewartson (DS) system. A classification of the type of DS system encountered is given, with regard to the value of the physical parameters (external field and wave frequency). Numerical computations show the various possible behaviours of the pulse. Blow-up and spreading out occur, as well as shape modifications. Interaction with electromagnetic long waves can even stabilize the pulse, or cut it into several parts

Electromagnetic waves in ferrites: from linear absorption to the nonlinear Schrödinger equation

We examine the effect of damping on the nonlinear modulation of an electromagnetic plane wave in a ferrite. Depending on the value of the damping constant, the time evolution of the amplitude of the wave is either a simple exponential decay, or is described either by a nonlinear Schrödinger (NLS) equation, or by a perturbed NLS equation. We give a new exact solution to this latter equation, and a way to compute approximate solutions