A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit

2018

Stud Appl Math

We present a detailed numerical study of various blow‐up issues in the context of the focusing Davey–Stewartson II equation. To this end, we study Gaussian initial data and perturbations of the lump and the explicit blow‐up solution due to Ozawa. Based on the numerical results it is conjectured that the blow‐up in all cases is self‐similar, and that the time‐dependent scaling behaves as in the Ozawa solution and not as in the stable blow‐up of standard L2 critical nonlinear Schrödinger equation. The blow‐up profile is given by a dynamically rescaled lump.

“…If one is interested in the solution to the DS II equation for localized initial data varying on length scales of order

1 / ε

, and this for times of order

1 / ε

with

ε ≪ 1

, one way to treat this is a change of coordinates

x \mapsto ε x

y \mapsto ε y

, and

t \mapsto ε t

. For , this leads to the equation (in an abuse of notation we use the same symbols as before)

\begin{matrix} i ε \partial_{t} ψ + ε^{2} \partial_{x x} ψ - ε^{2} \partial_{y y} ψ - 2 (() Φ + {| ψ |}^{2}) ψ & = & 0, \\ \partial_{x x} normalΦ + \partial_{y y} normalΦ + 2 \partial_{x x} {false| ψ false|}^{2} & = & 0 . \end{matrix}

Section: Numerical Resultsmentioning

confidence: 99%

Numerical Study of Blow‐Up Mechanisms for Davey–Stewartson II Systems

2018

Stud Appl Math

“…In this section we present a reformulation of system (2), which is suited for an efficient numerical treatment and discuss the employed numerical approach.…”

Section: Integral Equation and Numerical Approachesmentioning

confidence: 99%

“…Integral equations. The CGO solutions to (2) have an essential singularity at infinity which is numerically problematic for obvious reasons. The quantities Φ 1 and Φ 2 defined in (4), with asymptotic normalization (6), are well suited for numerical simulations.…”

Section: Integral Equation and Numerical Approachesmentioning

confidence: 99%

“…The reflection coefficient (3) was discussed in detail in [8], here the focus is on the solutions to the D-bar system (2). Nevertheless, we note here that it can be obtained from a given solution Φ 1 in a straight forward way at essentially no additional computational cost.…”

Section: Integral Equation and Numerical Approachesmentioning

confidence: 99%

“…(The numerical method is summarized in Section 2 below.) Subsequent to that, in [2], the asymptotic behavior of the DSII equation (1) with → 0 is considered, along with a number of different numerical methods aimed at elucidating the challenges in the asymptotic analysis.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectral approach to the scattering map for the semi-classical defocusing Davey–Stewartson II equation

McLaughlin²,

Physica D: Nonlinear Phenomena

2019

The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.

Large ∣k∣ Behavior of Complex Geometric Optics Solutions to d‐bar Problems

Sjöstrand

2022

Comm Pure Appl Math

Complex geometric optics solutions to a system of d‐bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey‐Stewartson II equations are studied for large values of the spectral parameter k. For potentials for some , it is shown that the solution converges as the geometric series in . For potentials q being the characteristic function of a strictly convex open set with smooth boundary, this still holds with s = 3/2, i.e., with instead of . The leading‐order contributions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.