Existence and stability of modulating pulse solutions in Maxwell’s equations describing nonlinear optics

Modulating pulse solutions play a big rôle in modern long distance high speed communication. Such solutions consist of a traveling pulse-like envelope modulating an underlying electromagnetic wave. In this paper we show that under certain assumptions such solutions exist and are dynamically stable for the associated nonlinear partial differential equations, namely Maxwells integro-differential equations describing nonlinear optics. The analysis is worked out in detail for bulk media, and we discuss how the results extend to optical fibers and to parametrically forced systems.     

Maxwell’s equations in inhomogeneous bi-anisotropic materials: Existence,uniqueness and stability for the initial value problem

In the present paper inhomogeneous bi-anisotropic materials characterized by matrices of electric permittivity, magnetic permeability and magnetoelectric characteristics are considered. All elements of these matrices are functions of the position in three dimensional space. The time-dependent Maxwell’s equations describe the electromagnetic wave propagation in these materials. Maxwell’s equations together with zero initial data are analyzed and a statement of the initial value problem (IVP) is formulated. This IVP is reduced to the IVP for a symmetric hyperbolic system of partial differential equations of the first order. Applying the theory of a symmetric hyperbolic system, new existence, uniqueness and stability estimate theorems have been obtained for the IVP of Maxwell’s equations in inhomogeneous bi-anisotropic materials.     

Darboux transformation and soliton solutions for the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics

In this paper, by virtue of the Darboux transformation (DT) and symbolic computation, the quintic generalization of the coupled cubic nonlinear Schrödinger equations from twin-core nonlinear optical fibers and waveguides are studied, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained and the corresponding DT is constructed. Moreover, one-, two- and three-soliton solutions are presented in the forms of modulus. Features of solitons are graphically discussed: (1) head-on and overtaking elastic collisions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) energy-exchanging collisions of the three solitons.     

A new FDTD scheme for Maxwell’s equations in Kerr-type nonlinear media

Numerical Algorithms - In this paper, we develop a totally new direct finite difference solver for solving the Maxwell’s equations in Kerr-type nonlinear media. The direct method is free of...     

Existence and uniqueness of solutions to singular ODE’s

We present an existence and uniqueness result for ordinary differential equations based upon the use of weighted spaces. We deal with singular and discontinuous right-hand-sides of the equation.     

About inclusion of Maxwell’s equations in systems relativistic of the invariant equations

Maxwell’s equations, relativistic invariant equations, foundations of difference schemes.     

Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.     

Thompson’s metric and global stability of difference equations

We apply the Thompson’s metric to study the global stability of the equilibium of the following difference equation

Birkhoff’s theorem and viscosity solutions of Hamilton-Jacobi equations

We obtain a partial generalization of Birkhoff’s theorem of invariant curve to higher dimesional case in the context of viscosity solutions of Hamilton-Jacobi equations, or weak KAM theory. This is a new approach after Herman’s proof. This work was supported by the National Basic Research Program of China (Grant No. 2007CB814800) and National Natural Science Foundation of China (Grant No. 10301012)     

Maple packages for computing Hirota’s bilinear equation and multisoliton solutions of nonlinear evolution equations

The Hirota method for generating Hirota’s bilinear equation and constructing soliton solutions of nonlinear evolution equations is discussed and illustrated. Two Maple programs Bilinearization and Multisoliton are presented to automatically calculate Hirota’s bilinear equations for nonlinear evolution equations and to compute their N-soliton solutions for N = 1, 2 or 3, respectively. Different kinds of examples are used to demonstrate the effectiveness of the packages.     

Stochastic-periodic homogenization of Maxwell’s equations with linear and periodic conductivity

Stochastic-periodic homogenization is studied for the Maxwell equations with the linear and periodic electric conductivity. It is shown by the stochastic-two-scale convergence method that the sequence of solutions to a class of highly oscillatory problems converges to the solution of a homogenized Maxwell equation.     

Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schrödinger equations

We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrödinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the Nehari manifold.Furthermore, we show that using the so-called fibering method and the Lusternik–Schnirel’man theory there exist infinitely many solutions, actually a countable family of critical points, for such a semilinear bi-harmonic Schrödinger system under study in this work.     

The stability problem and special solutions for the 5-components Maxwell–Bloch equations

For the 5-components Maxwell–Bloch system the stability problem for the isolated equilibria is completely solved. Using the geometry of the symplectic leaves, a detailed construction of the homoclinic orbits is given. Studying the problem of invariant sets for the system, we discover a rich family of periodic solutions in explicit form.