Analysis

Abstract

The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit.As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k.For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k D 0, which yields an asymptotic formula for the reflection coefficient at k D 0 and suggests an annular structure for the solution that may be exploited when k ¤ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as # 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory.

A note on the dependence of solutions on functional parameters for nonlinear Sturm-Liouville problems

“…Taking into account (2.2) and the boundedness of (u m ) m∈N in [0, 1], we infer the boundedness of (u m ) m∈N in W 1,2 (0, 1) and further, we state the existence of a subsequence of (u m ) m∈N (still denoted by (u m ) m∈N ) weakly convergent in W 1,2 (0, 1) to a certain u 0 ∈ W 1,2 (0, 1). The Rellich-Kondrashov theorem ( [12]) yields the uniform convergence of…”

“…In the wide literature devoted to BVPs similar to (1.1)-(1.3) (see e.g. [4][5][6][7][8][9][10][11][12][13][17][18][19][20][21] and references therein) the authors investigate mainly the existence of solutions for (1.1) under a variety of boundary conditions. Moreover, in the last fifty years, we could observe increasing interest in investigating sufficient conditions for the oscillation or nonoscillation of solutions of various classes of ODEs ( [1][2][3][5][6][7][8][9], and references therein).…”

Spectral Properties of Some Degenerate Elliptic Differential Operators

“…We may assume without loss of generality that δ * = δ ∈ (0, 1−β 2 ). It will suffice to prove the inequality for functions u ∈ C ∞ 0 (Ω) that are real-valued and nonnegative (Lieb & Loss [21], pp.176-177). For u ∈ C ∞ 0 (Ω) and u = d 1−β 2 v it follows from integrating by parts that and −∆d(x) ≥ 0 in Ω δ k for k sufficiently large and C(α, β) defined in (3.7).…”

“…In this paper, we will use the Sobolev space H 1 (G) = W 1,2 (G) for an open set G ⊂ R n , see Lieb and Loss [21], chapter 7.…”

Rise of Correlations of Transformation Strains in Random Polycrystals

“…For such a given admissible configuration ε T i , we assume a given imposed displacement that is chosen in the direction transversal to the laminates 1 : 13) together with the zero-traction boundary conditions…”