ATTRACTOR FOR WEAKLY DAMPED DRIVEN LONG WAVE-SHORT WAVE RESONANCE EQUATION

Abstract The existence of solution and universal attractor for weakly damped driven long wave short wave resonance equation are obtained.

SOME EXTENDED RESULTS OF“SUBHARMONIC RESONANCE BIFURCATION THEORY OF NONLINEAR MATHIEU EQUATION”

The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.

ON NECESSARY CONDITIONS FOR RESONANCE IN TURNING POINT PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

In this paper, some misunderstam igs concerning the necessary conditions for resonance for ordinary differential equations with turning point have been corrected, and a recursive process for finding the sequence of necessary conditions for resonance has beenoffered.

Bright soliton dynamics for resonant nonlinear Schrödinger equation with generalized cubic-quintic nonlinearity

For systems modeled by the resonant nonlinear Schrödinger equation(RNLSE)with generalized cubic-quintic nonlinearity,we derive the bright soliton solution of the equation in(1+1)dimensions,using the modified F-expansion method along with the novel ansatz of F-base function.Furthermore,we extend the analytical study of soliton dynamics to higher(2+1)and(3+1)dimensions by using the self-similar method,and demonstrate the soliton behavior via graphical illustration.Moreover,we investigate the effect of the resonance term on bright soliton solution in(1+1)dimensions.Additionally,we consider the nonlinear equation models with perturbation terms and derive the bright soliton solutions for the one-dimensional(1D)to three-dimensional(3D)cases.The theoretical results derived can be used to guide the experimental studies and observations of bright solitons in systems described by RNLSE model.

Application of coupled equation method on resonance processes of atomic lithium

The coupled equation method （CEM） has been applied to investigating the resonance structures for the ground state 1s^22s^ 2S of the neutral lithium from the first threshold up to 64.5 eV. Resonance structures of atomic lithium due to single excitations of the ls and 2s electrons are studied by infinite-order calculations in detail. The effect of spin-orbit splitting is also included for some of the low-lying ls2snp（↑↓） resonance, and the influence of the interference between 1s^2s^3 Snp .↓ and 1s2s^ 1 Snp ↑ states on the resonance structure has been confirmed theoretically. The results show that the presented technique can give the reasonable resonance structures very well in photoionization processes.

Relating Some Nonlinear Systems to a Cold Plasma Magnetoacoustic System

Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schrödinger equation.

Projective Dirichlet Boundary Condition with Applications to a Geometric Problem

Given a domain Ω R^n, let λ 〉 0 be an eigenvalue of the elliptic operator L ：= ∑i,j^n= 1δ/δxi on Ω for Dirichlet condition. For a function f ∈ L2（Ω）, it is known that the linear resonance equation Lu ＋ λu = f in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition Pλ（u｜δΩ） = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ｜｜u｜｜2,2 ≤ C（｜｜f｜｜2 ＋ ｜｜g｜｜2,2） under suitable regularity assumptions on δΩ and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W^2~＇P-estimates and the C^2,α-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean （Cartan） torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.