Xiaoping Xu
The Davey-Stewartson equations are used to describe the long time evolution of a three-dimensional packets of surface waves. Assuming that the argument functions are quadratic in spacial variables, we find in this paper various exact solutions modulo the most known symmetry transformations for the Davey-Stewartson equations.
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We consider a focusing Davey-Stewartson system and construct the solution of the Cauchy problem in the possible presence of exceptional points (and/or curves).
We consider the two dimensional focusing Davey-Stewartson II system and construct the global solution of the Cauchy problem for a dense in $L^2(\mathbb C)$ set of initial data. We do not assume that the initial data is small. So, the solutions may have singularities. We show that the blow-up may occur only on a real analytic variety and the variety is bounded in each strip $t \leq T$.
The study of 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 for large values of the spectral parameter $k$ in \cite{KlSjSt20} is extended to the reflection coefficient. For the case of potentials $q$ with compact support on some domain $Ω$ with smooth strictly convex boundary, improved asymptotic relations are provided.
Nonlinear dispersive partial differential equations such as the nonlinear Schrödinger equations can have solutions that blow-up. We numerically study the long time behavior and potential blowup of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa exact solutions. It is shown in this way that the lump is unstable to both blowup and dispersion, and that blowup in the Ozawa solution is generic.
We provide a numerical study of various issues pertaining to the dynamics of the Davey-Stewartson systems of the DS II type. In particular we investigate whether or not the properties (blow-up, radiation,...) displayed by the focusing and defocusing DS II integrable systems persist in the non integrable case.
We give a rigorous mathematical analysis of the one-soliton solution of the focusing Davey-Stewartson II equation and a proof of its instability under perturbation. Building on the fundamental perturbation analysis of Gadyl'shin and Kiselev, we verify their Assumption 1 and use Fredholm determinants to globalize their perturbation analysis.
This is a continuation of our previous paper arXiv:1904.07924, which is devoted to the construction of integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system; in this series of papers, we refer to a discretization of one of the two spatial variables as a semi-discretization. In this paper, we construct an integrable semi-discrete Davey-Stewartson system, which is essentially different from the semi-discrete Davey-Stewartson system proposed in the previous paper arXiv:1904.07924. We first obtain integrable semi-discretizations of the two el…
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 is as in the Ozawa solution and not as in the stable blow-up of standard $L^{2}$ critical nonlinear Schrödinger equations. The blow-up profile is given by a dynamically rescaled lump.
The dromion of the Davey-Stewartson-1 equation is studied under perturbation on the large time.
Dromions are exponentially localised coherent structures supported by nonlinear integrable evolution equations in two spatial dimensions.In the study of initial-value problems on the plane, such solutions occur only if one imposes nontrivial boundary conditions at infinity, a situation of dubious physical significance. However it is established here that dromions appear naturally in the study of boundary-value problems. In particular, it is shown that the long time asymptotics of the solution of the Davey-Stewartson I equation in the quarter plane with arbitrary initial conditions and with zer…
Equation with the lower negative time number in the Davey--Stewartson hierarchy
本文扩展了Davey-Stewartson层级中负时间变量数的构造方法，得到一个具有较低时间变量数的可积方程，并证明其时间约化系统是新的1+1维可积方程。
The integrable Davey-Stewartson system is a linear combination of the two elementary flows that commute: $\mathrm{i} q_{t_1} + q_{xx} + 2q\partial_y^{-1}\partial_x (|q|^2) =0$ and $\mathrm{i} q_{t_2} + q_{yy} + 2q\partial_x^{-1}\partial_y (|q|^2) =0$. In the literature, each elementary Davey-Stewartson flow is often called the Fokas system because it was studied by Fokas in the early 1990s. In fact, the integrability of the Davey-Stewartson system dates back to the work of Ablowitz and Haberman in 1975; the elementary Davey-Stewartson flows, as well as another integrable $(2+1)$-dimensional no…
We propose a natural (2+1)-dimensional generalization of the Ablowitz-Ladik lattice that is an integrable space discretization of the cubic nonlinear Schroedinger (NLS) system in 1+1 dimensions. By further requiring rotational symmetry of order 2 in the two-dimensional lattice, we identify an appropriate change of dependent variables, which translates the (2+1)-dimensional Ablowitz-Ladik lattice into a suitable space discretization of the Davey-Stewartson system. The space-discrete Davey-Stewartson system has a Lax pair and allows the complex conjugation reduction between two dependent variabl…
We consider the elliptic-elliptic, focussing Davey-Stewartson equations, which have an explicit bright line soliton solution. The existence of a family of periodic solitons, which have the profile of the line soliton in the longitudinal spatial direction and are periodic in the transverse spatial direction, is established using dynamical systems arguments. We also show that the line soliton is linearly unstable with respect to perturbations in the transverse direction.
A Modified Davey-Stewartson System of Nonlinear Dust Acoustic Waves in (3+1)-Dimensions: Lie Symmetries and Exact Solutions
本文研究(3+1)维修正Davey-Stewartson系统，通过变换消除常复数势项并刻画其李对称代数结构，进而获得线孤子、扭结孤子等精确解并分析行波解的稳定性。
We study the general (2+1)-dimensional Davey-Stewartson (DS) equations with nonlinear and gain terms and acquire explicit solutions through variable separation approach. In particular, we deduce some main novel excitations for the DS equations, and further demonstrate different features of these excitations. More importantly, the similar solutions and excitations can be predicted to exist in other related revolution equations such as nonlinear Schrödinger equation to explain the Bose-Einstein condensation.
We present the first detailed numerical study of the semiclassical limit of the Davey-Stewartson II equations both for the focusing and the defocusing variant. We concentrate on rapidly decreasing initial data with a single hump. The formal limit of these equations for vanishing semiclassical parameter $ε$, the semiclassical equations, are numerically integrated up to the formation of a shock. The use of parallelized algorithms allows to determine the critical time $t_{c}$ and the critical solution for these $2+1$-dimensional shocks. It is shown that the solutions generically break in isolated…
We show that any equation from the Davey--Stewartson hierarchy induces an infinite family of geometrically different deformations of tori in $\R^4$ preserving the Willmore functional. We expose a derivation of the Weierstrass representation for surfaces in the four-space which is not unique in difference from the case of surfaces in the three-space. This non-uniqueness implies that the spectral curve of a torus in $\R^4$ is not uniquely defined as a complex curve formed by the Floquet multipliers.
A family of higher-order rational lumps on non-zero constant background of Davey-Stewartson (DS) II equation are investigated. These solutions have multiple peaks whose heights and trajectories are approximately given by asymptotical analysis. It is found that the heights are time-dependent and for large time they approach the same constant height value of the first-order fundamental lump. The resulting trajectories are considered and it is found that the scattering angle can assume arbitrary values in the interval of $(\fracπ{2}, π)$ which is markedly distinct from the necessary orthogonal sc…
Integrable (3+1)-dimensional generalization for dispersionless Davey--Stewartson system
本文利用涉及接触向量场的 Lax pair 提出了一个 (3+1) 维无色散可积系统，作为 (2+1) 维无色散 Davey-Stewartson 系统的可积高维推广，为现代数学物理及后续研究奠定基础。
General dark solitons and mixed solutions consisting of dark solitons and breathers for the third-type Davey-Stewartson (DS-III) equation are derived by employing the bilinear method. By introducing the two differential operators, semi-rational solutions consisting of rogue waves, breathers and solitons are generated. These semi-rational solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. Under suitable parametric conditions, we derive general rogue wave solutions expressed in terms of rational functions. It is shown that the fundamental (simpl…
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 study 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 i…
We study the symmetry group properties of the variable coefficient Davey-Stewartson (vcDS) system. The Lie point symmetry algebra with a Kac-Moody-Virasoro (KMV) structure is shown to be isomorphic to that of the usual (constant coefficient) DS system if and only if the coefficients satisfy some conditions. These conditions turn out to coincide with those for the vcDS system to be transformable to the DS system by a point transformation. The equivalence group of the vcDS system is applied to pick out the integrable subsystems from a class of non-integrable ones. Additionally, the full symmetry…
In this work, we study the dynamics of rogue waves in the partially $\cal{PT}$-symmetric nonlocal Davey-Stewartson(DS) systems. Using the Darboux transformation method, general rogue waves in the partially $\cal{PT}$-symmetric nonlocal DS equations are derived. For the partially $\cal{PT}$-symmetric nonlocal DS-I equation, the solutions are obtained and expressed in term of determinants. For the partially $\cal{PT}$-symmetric DS-II equation, the solutions are represented as quasi-Gram determinants. It is shown that the fundamental rogue waves in these two systems are rational solutions which a…
The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of $1+1$ dimensional quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In analogy with the recently developed analytic theory of periodic AWs of the focusing NLS equation, in this paper we extend these results to a $2+1$ dimensional context, concentrating on the focusing Davey-Stewartson 2 (DS2) equation, an integrable $2+1$ dimensional generalization of the…
Combinatorial geometry of the 2D Toda lattice and Davey Stewartson equation
研究二维托达晶格和戴维 - 斯图尔特森方程，借助Kodama和Williams受热带几何启发的算法方法，通过全非负格拉斯曼流形元素生成等高线图，推广其方法并在渐近情况完善前人工作。
The periodic N breather anomalous wave solution of the Davey-Stewartson equations; first appearance, recurrence, and blow up properties
该研究构建了Davey-Stewartson方程的Akhmediev型N呼吸子反常波解，分析了其极限情形与爆破约束，并用匹配渐近展开法探讨了双周期柯西问题中的解相关性及双不稳定模式下的无普遍爆破特性。
Rigorous Melnikov analysis is accomplished for Davey-Stewartson II equation under singular perturbation. Unstable fiber theorem and center-stable manifold theorem are established. The fact that the unperturbed homoclinic orbit, obtained via a Darboux transformation, is a classical solution, leads to the conclusion that only local well-posedness is necessary for a Melnikov measurement.
The Davey Stewartson hierarchy will be developed based on a set of three matrix differential operators. These equations will act as evolution equations for different types of surface deformation in Euclidean four space. The Weierstrass representation for surfaces will be developed and its uniqueness up to gauge transformations will be reviewed. Applications of the hierarchy will be given with regard to generating deformations of surfaces, and it will be shown that the Willmore functional is preserved under this kind of deformation.
Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $β$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<β<\frac{1}{3}$. A classical vari…
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