Cube-tilings of ℝ n and nonlinear codesand nonlinear codes

Families of nonlattice tilings of ℝ ⁿ by unit cubes are constructed. These tilings are specializations of certain families of nonlinear codes overGF(2). These cube-tilings provide building blocks for the construction of cube-tilings such that no two cubes have a high-dimensional face in common. We construct cube-tilings of ℝ ⁿ such that no two cubes have a common face of dimension exceeding .     

Three nonlinear integrable couplings of the nonlinear Schrödinger equations

Based on two types of expanding Lie algebras of a Lie algebra G, three isospectral problems are designed. Under the framework of zero curvature equation, three nonlinear integrable couplings of the nonlinear Schröding equations are generated. With the help of variational identity, we get the Hamiltonian structure of one of them. Furthermore, we get the result that the hierarchy is also integrable in sense of Liouville.     

A characterization of nonlinear φ-accretive operators

Let E be a Banach space and E(–,] a proper lower semi-continuous convex function. The main purpose of this paper is to characterize those m-accretive operators AE x E that are also -accretive. This is done by using the semigroup S generated by -A, and by first establishing a necessary and sufficient condition for to be a Lyapunov function for S. We also obtain similar results for accretive operators that are not necessarily m-accretive, and deduce invariance and order-preserving criteria for nonlinear semigroups.This research was partially supported by the National Science Foundation under Grant MCS-8102086.     

Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term

The orbital stability of standing waves of nonlinear Schrödinger equations with a general nonlinear term is investigated in this paper. We study the corresponding minimizing problem with L ²-constraint: $$E_\alpha = \inf\left\{\frac{1}{2}\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx - \int\limits_{\mathbb{R}^N} F(|u|) dx; u \in H^1(\mathbb{R}^N), \|u\|_{L^2(\mathbb{R}^N)}^2=\alpha\right\}.$$ We discuss when a minimizing sequence with respect to E _α is precompact. We prove that there exists α ₀ ≥ 0 such that there exists a global minimizer if α > α ₀ and there exists no global minimizer if α < α ₀. Moreover, some almost critical conditions which determine α ₀ = 0 or α ₀ > 0 are established, and the existence results with respect to ${E_{\alpha_0}}$ under some conditions are obtained.     

Normalized solutions of nonlinear Schrödinger equations

We consider the problem $$\left\{\begin{array}{ll}-\Delta u - g(u) = \lambda u,\\ u \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} u^2 = 1, \lambda \in \mathbb{R},\end{array}\right.$$ in dimension N ≥ 2. Here g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the L ²-unit sphere, and we show the existence of infinitely many solutions.     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu f(u) =0 in Ω, u=0 on Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u~3,u~2(u-p),u~2(u~2 -p),     

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.     

Classical solutions of nonlinear Schrödinger equations

We prove the existence of global classical solutions to the initial value problem for the nonlinear Schrödinger equation, iu_t–u+q(|u|²)u=0 in iu_t - u + (|u|²)u = in (t, x)xⁿ for 6n11.     

Analysis of search-extension method for finding multiple solutions of nonlinear problem

For numerical computations of multiple solutions of the nonlinear elliptic problemΔu f(u)=0 inΩ, u=0 onΓ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u∈H~(1 α),α>0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.     

Multiplicity of positive solutions of a nonlinear Schrödinger equation

We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: where if N3 and p(1, ) if N=1,2, and a(x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well := int a^–1(0) consisting of k components and the first eigenvalues of –+b(x) on _j under Dirichlet boundary condition are positive for all . Under these conditions we show that (PM) has at least 2^k–1 positive solutions for large . More precisely we show that for any given non-empty subset , (P) has a positive solutions u(x) for large . In addition for any sequence _n we can extract a subsequence _ni along which u_ni converges strongly in H¹(R^N). Moreover the limit function u(x)=lim_iu_ni satisfies (i) For jJ the restriction u|_j of u(x) to _j is a least energy solution of –v+b(x)v=v^p in _j and v=0 on _j. (ii) u(x)=0 for .Mathematics Subject Classifications (2000):35Q55, 35J20     

Hydrodynamic limits of the nonlinear Klein–Gordon equation

We perform the mathematical derivation of the compressible and incompressible Euler equations from the modulated nonlinear Klein–Gordon equation. Before the formation of singularities in the limit system, the nonrelativistic-semiclassical limit is shown to be the compressible Euler equations. If we further rescale the time variable, then in the semiclassical limit (the light speed kept fixed), the incompressible Euler equations are recovered. The proof involves the modulated energy introduced by Brenier (2000) [1].     

On a class of nonlinear Schrödinger equations

This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation:

Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations

In this paper we are concerned with multi-lump bound states of the nonlinear Schr?dinger equation for sufficiently small , where for and for . V is bounded on . For any finite collection of nondegenerate critical points of V, we show the uniqueness of solutions of the form for , where u is positive on and is a small perturbation of a sum of one-lump solutions concentrated near , respectively for sufficiently small . Received: 30 October 2001; in final form: 10 June 2002 /Published online: 2 December 2002 RID="*" ID="*" Research supported by Alexander von Humboldt Foundation in Germany and NSFC in China     

Autoresonance excitation of a soliton of the nonlinear Schrödinger equation

The action of an external parametric perturbation with slowly changing frequency on a soliton of the nonlinear Schrödinger equation is studied. Equations for the time evolution of the parameters of the perturbed soliton are derived. Conditions for the soliton phase locking are found, which relate the rate of change of the perturbation frequency, its amplitude, the wave number, and the phase to the initial data of the soliton. The cases when the initial amplitude of the soliton is small and when the amplitude of the soliton is of the order of unity are considered.