Interaction Effect of Plane Electromagnetic Waves in the Born–Infeld Nonlinear Electrodynamics

We calculate the interaction of plane electromagnetic waves in accordance with the Born–Infeld nonlinear electrodynamics in the vacuum segment of a ring laser. We show that at the present level of development in laser physics, it is possible not only to measure the direct impact of the quantum vacuum polarization effect on the generation frequencies of weak electromagnetic waves propagating in opposite directions in a ring laser but also to estimate the contribution of the Born–Infeld nonlinear electrodynamics to this effect.     

Erratum to: Stability of Diffusion Coefficients in an Inverse Problem for the Lotka-Volterra Competition System

First we establish a Carleman estimate for Lotka-Volterra competition-diffusion system of three equations with variable coefficients. Then the internal observations with two measurements are allowed to obtain the stability result for the inverse problem consisting of retrieving two smooth diffusion coefficients in the given parabolic system for the dimension n≤3. The proof of the results rely on Carleman estimates and certain energy estimates for parabolic system.     

The Global Solution for One Class of the System of LS Nonlinear Wave Interaction

Recently Djordjevie and Redekopp [1] found that the evolution equations describing the resonance interaction between the long wave and the short wave can be written as     

Erratum to: A Nonlinear Generalization of Singular Value Decomposition and Its Applications to Mathematical Modeling and Chaotic Cryptanalysis

Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively applied to Signal Processing, Statistical Analysis and Mathematical Modeling. We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. The method is to augment the embedding matrix with additional nonlinear columns derived from the initial embedding vectors and extract the nonlinear relationship using SVD. The paper demonstrates an application of nonlinear SVD to identify parameters when the signal is generated by a nonlinear transformation. Examples of maps (Logistic map and Henon map) and flows (Van der Pol oscillator and Duffing oscillator) are used to illustrate the method of nonlinear SVD to identify parameters. The paper presents the recovery of parameters in the following scenarios: (i) data generated by maps and flows, (ii) comparison of the method for both noisy and noise-free data, (iii) surrogate data analysis for both the noisy and noise-free cases. The paper includes two applications of the method: (i) Mathematical Modeling and (ii) Chaotic Cryptanalysis.     

Existence and Stability of Solitary Waves of an M-Coupled Nonlinear Schrödinger System

In this paper, the existence and stability results for ground state solutions of an m-coupled nonlinear Schrödinger system $$i\frac{∂}{∂ t}u_j+\frac{∂²}{∂x²}u_j+\sum\limits^m_{i=1}b_{ij}|u_i|^p|u_j|^{p-2}u_j=0,$$ are established, where $2 ≤ m, 2≤p<3$ and $u_j$ are complex-valued functions of $(x,t) ∈ \mathbb{R}^2, j=1,...,m$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}$. In contrast with other methods used before to establish existence and stability of solitary wave solutions where the constraints of the variational minimization problem are related to one another, our approach here characterizes ground state solutions as minimizers of an energy functional subject to independent constraints. The set of minimizers is shown to be orbitally stable and further information about the structure of the set is given in certain cases.     

General Decay of Solutions of a Nonlinear System of Viscoelastic Equations

In this paper we consider a system of two coupled viscoelastic equations with Dirichlet boundary condition which describes the interaction between two different fields arising in viscoelasticity. For certain class of relaxation functions and certain initial data, we prove that the decay rate of the solution energy is similar to that of relaxation functions which is not necessarily of exponential or polynomial type. This result improves earlier one of Messaoudi and Tatar (Appl. Anal. 87(3):247–263, 2008) and extends some existing results concerning the general decay for a single equation to the case of a system.     

Stability of Solitary Waves for the Nonlinear Derivative Schrodinger Equation

1 IntroductionConsider tke following nonlinear quintic derlvative Schrodinger equation u_t=iu_(xx)+ig(|u|~2)u+[(S_0+S_2|u|~2)u]_x,x∈R (1)with g(|u|~2)=c_3|u|~2+c_5|u|~4 (2)and s_0,s_2,c3,c_5 real constants Problem(1)-(2)appears in various physical applications,such as plasma physics,nonlinear optics,and nonrelativistic quantum physics. For the case s_0=s_w=0,equ.(1)becomes the classical nonlinear Schrodinger e-quation,which has been studied by many authors,especially for the stability ofsolitary waves.In this case(1)can be written in the following abstract Hamil-     

Standing Waves of the Coupled Nonlinear Schr ¨odinger Equations

In this paper, we study the existence of standIng waves of the coupled nonlinear Schrodinger equations. The proofs of which rely on the Lyapunov-Schmidt methods and contraction mapping principle are due to F. Weinstein in .     

Standing Waves of the Coupled Nonlinear Schrdinger Equations

In this paper, we study the existence of standing waves of the coupled nonlinear Schr dinger equationsThe proofs of which rely on the Lyapunov-Schmidt methods and contraction mapping principle are due to FWeinstein in [1].     

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

In the first part of these notes, we deal with first order Hamiltonian systems in the form where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian is required to be invariant with respect to the action of a group of isometries where is skew-symmetric and JA  = AJ. A standing wave is a solution having the form for some and such that . Given a solution of this type, it is natural to investigate its stability with respect to perturbations of the initial condition. In this context, the appropriate notion of stability is orbital stability in the usual sense for a dynamical system. We present some of the important criteria for establishing orbital stability of standing waves. In the second part we consider the nonlinear Schr?dinger equation which provides an interesting example of this situation where standing waves appear as time-harmonic solutions. We show how the general theory applies to this case and review what is known about stability. Received: January 2008     

Orbital Instability of Standing Waves for the Klein-Gordon-Zakharov System

O Introduction We consider the orbital instability of standing waves for the Klein-Gordon-Zakharov system with different propagation speeds in three space dimensions     

Erratum to: Risk-Sensitive Control with Near Monotone Cost

The infinite horizon risk-sensitive control problem for non-degenerate controlled diffusions is analyzed under a ‘near monotonicity’ condition on the running cost that penalizes large excursions of the process.     

On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation

In this paper, a higher-order method for the solution of a nonlinear scalar equation is presented. It is proved that the new method is locally convergent with an order of (m+2), where m is the highest order derivative used in the iterative formula. Some numerical examples are used to demonstrate the new method.     

Erratum to: A Variational Approach to Givental’s Nonlinear Maslov Index

In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental’s nonlinear Maslov index. As an application we prove a Bott–Samelson type obstruction theorem for positive loops of contactomorphisms.     

Integration Of A Higher-Order Nonlinear SchröDinger System with a Self-Consistent Source in the Class of Periodic Functions

Theoretical and Mathematical Physics - We use the inverse spectral method to integrate a higher-order nonlinear Schrödinger system with a self-consistent source in the class of periodic...