Mengxue Bai, Jian Zhang, Shihui Zhu
This paper is concerned with the generalized Davey-Stewarston system in two dimensional space. Existence and stability of small solitons are proved by solving two correlative constrained variational problems and spectrum analysis. In addition, multi-solitons with different speeds are constructed by bootstrap argument.
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Dromion perturbation for the Davey-Stewartson-1 equations
研究了Davey-Stewartson-1方程dromion的长时间扰动。
Explode-decay dromions in the non-isospectral Davey-Stewartson I (DSI) equation
在这封信中，我们报告了一种新型的爆炸衰变dromion的存在，这是指数局域的相干结构，其振幅随时间变化，通过Hirota方法的非等谱Davey-Stewartson方程我最近讨论的江。利用适当的变换，我们还指出，这样的解决方案也存在的等谱Davey-Stewartson我方程本身的电位仔细选择。
We consider the elliptic-elliptic Davey-Stewartson system in the three-dimensional Euclidean space, and we give sufficient conditions for the existence of finite time blow-up solutions in non-isotropic spaces. The proof is based on some general results on distributions defined via homogeneous symbols, in conjunction with a convexity argument.
The Davey Stewartson system in Weak $L^p$ Spaces
研究了$\re^n,\ n=2,3$中Davey-Stewartson系统的整体Cauchy问题.在弱$L^p$空间中，证明了小数据情形下解的存在唯一性.我们应用一个内插定理和推广的薛定谔方程推导出\cite{CVeVi}的Schr hartz估计。因此，我们得到自相似的解决方案。
A lower bound on the blow-up rate for the Davey-Stewartson system on the torus
本文研究了二维环面上具有立方非线性项的Davey-Stewartson系统的双曲-椭圆形式。在$H^s, 1/2 < s < 1$中，可以自然地设置用于研究该方程的爆破解。在本文中，我们证明了一个下界的爆破率为这些图。
Mass Concentration for the Davey-Stewartson System
本文研究了椭圆-椭圆型Davey-Stewartson方程组在描述水面波演化过程中的爆破解。我们证明了一个质量浓度性质的$H^{1}$ -解决方案，类似于一个已知的$L^{2}$ -临界非线性薛定谔方程。我们还证明了$L^{2}$ -解决方案的质量浓度的结果。
On the analytical approach to the N-fold Bäcklund transformation of Davey-Stewartson equation
利用Lax本征函数在复本征值平面上的解析结构，构造了Davey-Stewartson方程的N重Bäcklund变换.对于给定的N值，可以得到显式公式。最后给出了广义孤子解是如何由平凡孤子解生成的。
In this paper we study the regularity properties of solutions to the Davey-Stewartson system. It is shown that for initial data in a Sobolev space, the nonlinear part of the solution flow resides in a smoother space than the initial data for all times. We also obtain that the Sobolev norm of the nonlinear part of the evolution grows at most polynomially. As an application of the smoothing estimate, we study the long term dynamics of the forced and weakly damped Davey-Stewartson system. In this regard we give a new proof for the existence and smoothness of the global attractors in the energy sp…
Geometric optics and instability for NLS and Davey-Stewartson models
研究了具有规范不变幂律非线性项和非局域扰动的多维非线性薛定谔方程的慢调制高频波相互作用。该模型包括Davey-Stewartson系统的椭圆-椭圆和双曲-椭圆变体。我们的分析揭示了一个新的本地化现象的非局部扰动的高频政权，并允许我们推断出强烈的不稳定性的结果柯西问题的负序索伯列夫空间，在这里我们证明了规范膨胀与无限的正则性损失的建设性方法。
We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey-Stewartson II equation. We then use it to prove global well-posedness and scattering in $L^2$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension $2$, for conductivities $σ>0$ with $\log σ\in \dot H^1$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $L^2$-boundedness of pseudo-dif…
On the Moutard transformation singularity for the Davey--Stewartson II equation
本文研究 Davey–Stewartson II 方程，给出构造含两个不定点的 Moutard 变换奇性的方法，并据此解释任意阶“质量下降”现象。
We prove existence of small-amplitude modulated solitary waves for the full-dispersion Kadomtsev--Petviashvilii (FDKP) equation with weak surface tension. The resulting waves are small-order perturbations of scaled, translated and frequency-shifted solutions of a Davey--Stewartson (DS) type equation. The construction is variational and relies upon a series of reductive steps which transform the FDKP functional to a perturbed scaling of the DS functional, for which least-energy ground states are found. We also establish a convergence result showing that scalings of FDKP solitary waves converge …
We show that the inverse scattering map for the linear system associated with the defocussing Davey-Stewartson II equation is locally Lipschitz continuous with locally Lipschitz continuous inverse on $H^{1,1}(R^2)$. From the inverse scattering method we then obtain global well-posedness for the defocussing Davey-Stewartson II equation. We show that these global solutions are dispersive by computing their leading asymptotic behavior as $t \rightarrow \infty$ in terms of an associated linear problem.
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).
An efficient high precision hybrid numerical approach for integrable Davey-Stewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study of DS I solutions in this class. Localized stationary solutions are constructed and shown to be unstable against dispersion and blow-up. A finite-time blow-up of initial data in the Schwartz class of smooth rapidly decreasing functions is discussed.
Lie symmetry algebra of the dispersionless Davey-Stewartson (dDS) system is shown to be infinite-dimensional. The structure of the algebra turns out to be Kac-Moody-Virasoro one, which is typical for integrable evolution equations in $2+1$-dimensions. Symmetry group transformations are constructed using a direct (global) approach. They are split into both connected and discrete ones. Several exact solutions are obtained as an application of the symmetry properties.
Quadratic-Argument Approach to the Davey-Stewartson Equations
本文用Davey-Stewartson方程描述三维表面波包的长时间演化。假设自变量函数在空间变量中是二次的，本文找到了Davey-Stewartson方程的模对称变换的各种精确解。
A numerical approach to Blow-up issues for Davey-Stewartson II type systems
我们提供了一个数值研究有关的DS II型的Davey-Stewartson系统的动力学的各种问题。特别是，我们调查是否属性（爆破，辐射，.）DS II可积系统在不可积情况下仍然存在。
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.
In this paper, we prove global well-posedness and scattering of the Cauchy problem for the elliptic-elliptic Davey-Stewartson system (eeDS) for initial data $u_{0}\in L^{2}(\mathbb{R}^{2})$ in the defocusing case and for $u_{0}\in L^{2}(\mathbb{R}^{2})$ with mass below that of the ground state in the focusing case. This result resolves the large data problem at the scaling-critical regularity left open by Ghidaglia and Saut in their work initiating the mathematical study of the Cauchy problem for the system. Our proof uses the concentration compactness/rigidity road map of Kenig and Merle toge…
The Ozawa solution to the Davey--Stewartson II equations and surface theory
该研究从曲面理论视角研究Davey--Stewartson II方程的Ozawa解，提出由其支配的曲面孤子形变，明确了解爆破时刻对应形变曲面的奇异性。
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.
Numerical Study of Blowup in the Davey-Stewartson System
非线性色散偏微分方程，如非线性薛定谔方程，可以有解爆破。通过对集中解和Ozawa精确解的扰动分析，数值研究了聚焦Davey-Stewartson II方程解的长时间行为和可能的爆破.它表明，在这种方式下，块是不稳定的爆破和分散，爆破在小泽解决方案是通用的。
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.
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 Davey-Stewartson I Equation on the Quarter Plane with Homogeneous Dirichlet Boundary Conditions
Dromion是二维空间中由非线性可积演化方程支持的指数局域相干结构。在平面上的初值问题研究中，只有当在无穷远处施加非平凡边界条件时才会出现这样的解，这种情况的物理意义是可疑的。然而，它是建立在这里，dromion自然出现在边值问题的研究。特别地，证明了具有任意初始条件和零Dirichlet边界条件的Davey-Stewartson I方程在四分之一平面上的解的长时间渐近性是由dromion支配的.非零Dirichlet边界条件的情况也进行了讨论。
In the present study we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in variety of parameter regimes under which the finite blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the $α$-models of turbulence and therefore are called th…
Numerical Study of the semiclassical limit of the Davey-Stewartson II equations
我们提出了第一个详细的数值研究的半经典极限的Davey-Stewartson II方程的聚焦和散焦的变体。我们专注于快速减少的初始数据与一个单一的驼峰。这些方程的形式限制为零的半经典参数$ε$，半经典方程，数值积分的冲击波的形成。并行算法的使用允许确定这些$2+1$维冲击的临界时间$t_{c}$和临界解。结果表明，解一般在孤立点处破裂，类似于$1+1$维立方非线性薛定谔方程的情况，即，散焦情况下的立方奇点和聚焦情况下的平方根奇点。对于小的$ε$值，完整的Davey-Stewartson II方程被集成为相同的初始数据直到临界时间$t_{c}$。这些解决方案之间的差异在$ε$的缩放被发现是相同的$1+1$维的情况下，成比例的$ε^{2/7}$的散焦的情况下，成比例的$ε^{2/5}$的聚焦的情况下。我们记录的Davey-Stewartson II解决方案的小$ε$的时间远远大于临界时间$t_{c}$。结果表明，快速调制振荡区附近的半经典方程的解决方案的冲击形成。对于较小的$ε$，振荡区变得更小并且更清晰地界定为透镜形区域。快速振荡也被发现在聚焦的情况下，初始数据的奇异性的解决方案的半经典方程不一致。
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…
On Solvability of the Cauchy problem for the partial differential equations
证明了Davey-Stewartson-II型方程组初边值问题的经典可解性。
We consider a scattering map that arises in the $\bar \partial $-approach to the scattering theory for the Davey-Stewartson II equation and show that the map is an invertible map between certain weighted $L^2$ Sobolev spaces.
These notes are a considerably revised and expanded version of expository lectures given at the Fields Institute Workshop on "Nonlinear Dispersive Partial Differential Equations and Inverse Scattering" in August 2017. We give a complete and self-contained treatment of inverse scattering for the defocussing cubic NLS in one-dimension, following the 2003 paper of Deift and Zhou, and the defocussing Davey-Stewartson equation in two space dimensions, following the work of Perry and more recent work of Nachman, Regev, and Tataru.
Melnikov Analysis for Singularly Perturbed DSII Equation
对Davey-Stewartson II方程在奇异扰动下进行了严格的Melnikov分析.建立了不稳定纤维定理和中心稳定流形定理。通过达布变换得到的未扰动同宿轨道是一个经典解，这一事实导致了只有局部适定性对于Melnikov测量是必要的结论。
Breathers and solitons of generalized nonlinear Schrödinger equations as degenerations of algebro-geometric solutions
我们给出了多分量非线性薛定谔方程的初等函数形式的新解和Davey-Stewartson方程的已知解，如多孤子解、呼吸子解、dromion解和块解.这些解决方案给出了一个简单的行列式形式，并获得了限制情况下，在适当的退化以前派生的代数几何解决方案。特别地，我们首次给出了多分量非线性薛定谔方程的呼吸子解和有理呼吸子解。
We consider the $\overline{\partial}$-Dirac system that Ablowitz and Fokas used to transform the defocussing Davey-Stewartson system to a linear evolution equation. The nonlinear Plancherel identity for the associated scattering transform was established by Beals and Coifman for Schwartz functions. Sung extended the validity of the identity to functions belonging to $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ and Brown to $L^2(\mathbb{R}^2)$-functions with sufficiently small norm. More recently, Perry extended to the weighted Sobolev space $H^{1,1}(\mathbb{R}^2)$ and here we extend to $H^{s…
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