C. Özemir
This article is devoted to discovering Lie symmetry algebra of a (3+1)-dimensional Davey-Stewartson system which appears in the field of plasma physics. It is found that the algebra is an infinite dimensional one and of Kac-Moody type. Making use of these symmetries, some reduced equations to lower dimensions are also presented.
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We use example of the Davey-Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.
The dDS (dispersionless Davey-Stewartson) hierarchy is constructed by two eigenfunctions of a special vector field. This hierarchy consists the infinite symmetries of the dDS system. Further, this paper explores the tau function, the Riemann-Hilbert problem and Hamilton-Jacobi theory related to dDS hierarchy.
The infinite many symmetries of Davey-Stewartson (DS) system are closely connected to the integrable deformations of surfaces in a four-dimensional space. In this paper, we give a direct algorithm to construct the expression of the DS hierarchy by two scalar pseudo-differential operators involving partial derivatives.
Dromion solutions of noncommutative Davey-Stewartson equations
考虑Davey-Stewartson方程的非对易形式，通过Darboux变换和二元达布变换得到两类拟行列式解.这些解决方案可以通过直接替换来验证。然后，我们计算方程的dromion解决方案，并获得计算机绘图在非交换设置。
In this paper, we continue the study of the Davey-Stewartson system which is one of the most important$(2+1)$ dimensional integrable models. As we showed in the previous paper, the dDS (dispersionless Davey-Stewartson) system arises from Hamiltonian vector fields Lax pair. A new hierarchy of compatible PDEs defining infinitely many symmetries, which is associated with the dDS system, is defined in this paper. We show that this hierarchy arises from the commutation condition of a particular series of one-parameter Hamiltonian vector fields.
On Classification of Integrable Davey-Stewartson Type Equations
本文主要研究可积Davey-Stewartson型方程的分类问题。潜在的可变形的色散系统的列表是通过这样的系统必须由多项式色散拉克斯对产生的要求获得的。基于流体动力学约化方法的微扰方法被用来恢复可积系统及其拉克斯对。有些发现的系统似乎是新的。
In this paper, the semiclassical limit of Davey-Stewartson systems are studied. It shows that these dispersionless limited integrable systems of hydrodynamic type, which are defined as dDS (dispersionless Davey-Stewartson) systems, are arisen from the commutation condition of Lax pairs of one-parameter vector fields. The relevant nonlinear Riemann-Hilbert problems with some symmetries for the dDS-II system are also constructed. This kind of Riemann-Hilbert problems are meaningful for applying the formal inverse scattering transform method, recently developed by Manakov and Santini, to study th…
Relation between hyperbolic Nizhnik-Novikov-Veselov equation and stationary Davey-Stewartson II equation
本文提出一个三元Lax系统，其中两个方程构成定常Davey-Stewartson II方程的拉克斯对。在一定的非线性约束下，该Lax系统的完全可积条件包含双曲型Nizhnik-Novikov-Veselov方程及其标准拉克斯对.利用Davey-Stewartson II方程的达布变换求解双曲型Nizhnik-Novikov-Veselov方程。利用达布变换，得到了整体$n$孤子解.证明了每个$n$ -孤子解在空间无穷远处一致地指数趋近于零，并且在时间无穷远处渐近于$n^2$峰丛。
Finite-dimensional integrable systems associated with Davey-Stewartson I equation
对于1+2维可积的Davey-Stewartson I方程，我们已经通过非线性约束找到了它的1+1维形式的拉克斯对.本文对这个1+1维系统进行二次非线性化，得到三个带Neumann型约束的1+0维Hamilton系统。得到了对合守恒积分的全集，并证明了它们的函数独立性。因此，Hamilton系统在Liouville意义下是完全可积的。以求解这些经典的Hamilton系统为例，得到了Davey-Stewartson I方程的周期解.
The generalized Davey-Stewartson equations, its Kac-Moody-Virasoro symmetry algebra and relation to DS equations
我们计算了广义Davey-Stewartson（GDS）方程的李对称代数，证明了在一定的参数条件下，广义Davey-Stewartson方程是无穷维的，并且与标准可积Davey-Stewartson方程同构，后者具有特定的Kac-Moody-Virasoro圈代数结构.我们讨论了如何Virasoro部分的对称代数可以用来构造新的解决方案，这是至关重要的证明存在的爆破配置文件，从已知的使用李群的三维子代数，即$\Sl(2,\mathbb{R})$产生的变换。我们进一步讨论GDS方程的可积性方面。
On the construction of solutions of the Davey--Stewartson I equation using an open Toda chain
本文提出基于dressing链（利用$A_2$广义Toda格）构造Davey--Stewartson I方程显式解的方法，得到含两个任意函数的新解类。
Darboux transformations and global explicit solutions for nonlocal Davey-Stewartson I equation
对于非局部Davey-Stewartson I方程，考虑了达布变换，得到了解的显式表达式.像1+1维的非局部方程一样，许多解可能有奇点。然而，通过适当地选择拉克斯对解中的参数，证明了由分别为零和$t$的指数函数的种子解通过$n$次达布变换得到的解是非局部Davey-Stewartson I方程的整体解.当种子解为零时，导出的解是孤子解，即它们是有界的且具有$n$峰;当种子解是$t$的指数函数时，导出的解是“线暗孤子”解，即它们是有界的且它们的范数沿某些直线快速变化.
On lump instability of Davey--Stewartson II equation
我们表明，团块（孤子）的Davey-Stewartson II方程的初始数据的小扰动下失败。
Comment on "An optimal system, invariant solutions, conservation laws, and complete classification of Lie group symmetries for a generalized (2+1)-dim. Davey-Stewartson system of equations for the wave propagation in water of finite depth"
本文指出近期一篇关于广义(2+1)维Davey-Stewartson方程组的论文存在重大错误、遗漏和不准确之处。
The Fourier method for the linearized Davey-Stewartson I equation
用Fourier方法求解变系数线性化Davey-Stewartson方程。该方法使用的逆散射变换的Davey-Stewartson方程。
Conformal geometry of the (discrete) Schwarzian Davey-Stewartson II hierarchy
研究了Schwarzian Davey-Stewartson II族及其离散类似物的共形几何。与离散和连续等温表面和广义Clifford配置的连接记录。给出了Schwarzian Davey-Stewartson II流作为共形浸入曲面的可积变形的解释。
A prototypical example of a rogue wave structure in a two-dimensional model is presented in the context of the Davey-Stewartson~II (DS~II) equation arising in water waves. The analytical methodology involves a Taylor expansion of an eigenfunctionof the model's Lax pair which is used to form a hierarchy of infinitely many new eigenfunctions. These are used for the construction of two-dimensional (2D) rogue waves (RWs) of the DS~II equation by the even-fold Darboux transformation (DT). The obtained 2D RWs, which are localized in both space and time, can be viewed as a 2D analogue of the Peregrin…
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。
Rogue curves in the Davey-Stewartson I equation
本文在 Davey-Stewartson I 方程中发现了波峰呈闭合或开放曲线的新型怪浪曲线，揭示了其在双线性表达式内部参数为实数且较大时产生，并通过双实变量多项式的根曲线进行了解析预测与验证。
The Moutard transform is constructed for the solutions of the Davey-Stewartson II equation. It is geometrically interpreted using the spinor (Weierstrass) representation of surfaces in four-dimensional Euclidean space. Using the Moutard transformation and minimal surfaces, examples of solutions are constructed that have smooth, rapidly decreasing initial data and lose their regularity in a finite time.
Solution of the initial value problem for the focusing Davey-Stewartson II system
我们考虑聚焦Davey-Stewartson系统，并在可能存在例外点（和/或曲线）的情况下构造柯西问题的解。
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。
On a discrete Davey-Stewartson system
在${\mathcal R}^1\times {\mathcal Z}^2$中提出了一个微分差分方程，并利用Hirota双线性方法对其进行了研究.这个方程有一个奇异的连续极限到一个系统，允许减少到Davey-Stewartson方程。这个离散DS系统的解的特征在于Casorati和Grammian行列式。基于此离散DS系统的双线性形式，我们构造了双线性贝克隆变换，使我们能够得到它的拉克斯对.
Rational and semi-rational solutions of the nonlocal Davey-Stewartson equations
本文将关于$x$的部分时间（$PT$）对称非局部Davey-Stewartson（DS）方程称为$x$ -非局部DS方程，而将完全$PT$对称非局部DSII方程称为非局部DSII方程.利用双线性方法得到了这些非局部DS方程的呼吸子、有理解和半有理解。对于$x$ -非局部DS方程，通常的（$2+1$）维呼吸子在$x$方向是周期的，在$y$方向是局域的.非奇异有理解是块，半有理解是由块、呼吸子和周期线波组成。对于非局部DSII方程，线呼吸子在$x$和$y$方向上都是周期性的，在剖面上具有平行线，但在时间上是局部的。非奇异有理解是（$2+1$）维线流氓波，它从一个恒定的背景中产生，消失在同一个恒定的背景中，这个过程只持续很短的时间。半理性的解决方案描述了线流氓波和周期线波的相互作用。
Asymptotics of perturbed soliton for Davey--Stewartson II equation
结果表明，在Davey-Stewartson（DS-Ⅱ）方程的块（孤子）的小扰动下，散射数据得到非孤子结构。因此，该解具有傅里叶型积分的形式。渐近分析表明，尽管存在色散，但解的渐近展开式的主项在大时间内仍具有孤波形式。
Asymptotics of soliton solution for the perturbed Davey-Stewartson-1 equations
研究了Davey-Stewartson-1方程在大时间扰动下的dromion。
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…
Dynamics of rogue waves in the Davey-Stewartson II equation
用双线性方法导出了Davey-Stewartson-II方程中的一般异常波，并通过行列式给出了其解。它表明，最简单的（基本）流氓波是线流氓波，它产生于恒定的背景线剖面，然后撤退到恒定的背景再次。它还表明，多流氓波描述了几个基本流氓波之间的相互作用，和高阶流氓波表现出不同的动力学（如从恒定的背景上升，但不撤退回它）。这些流氓波的一个显著特征是，在某些参数条件下，这些流氓波可以在有限时间内在孤立的空间点上爆炸到无穷大，即，在Davey-Stewartson-II方程中存在爆炸的异常波。
The Davey Stewartson system and the Bäcklund Transformations
本文考虑（耦合）Davey-Stewartson（DS）系统及其贝克隆变换（BT）.建立了DS系统、双Kadomtsev-Petviashvili（KP）系统和Ablowitz-Ladik层次（ALH）之间的关系。DS层次和双KP系统是等价的。ALH是DS系统的BT在一定程度上的还原。从耦合DS系统的BT出发，得到了新的耦合导数非线性薛定谔方程。
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。
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。
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.
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…
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。
On a (2+1)-dimensional generalization of the Ablowitz-Ladik lattice and a discrete Davey-Stewartson system
我们提出了一个自然的（2+1）维推广的Ablowitz-Ladik格，这是一个可积空间离散的立方非线性薛定谔（NLS）系统在1+1维。通过进一步要求二维晶格中的2阶旋转对称性，我们确定了一个适当的变化的因变量，这将（2+1）维Ablowitz-Ladik晶格转换成一个合适的空间离散的Davey-Stewartson系统。空间离散的Davey-Stewartson系统有一个拉克斯对，并允许在连续的情况下，两个因变量之间的复共轭约化。此外，它相对于空间反射理想对称。利用Hirota双线性方法，我们构造了一些精确解，如多参数解。
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.
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。
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。
Spectral approach to the scattering map for the semi-classical defocusing Davey-Stewartson II equation
本文提出了一种数值方法，用于解决半经典D-bar问题，通过将D-bar问题转化为复线性二次积分方程，并利用离散傅里叶变换和解析计算进行正则化，从而求解得到代数方程。
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…
Rogue waves in the Davey-Stewartson equation
本文用双线性方法导出了Davey-Stewartson-I方程中的一般异常波。它表明，最简单的（基本）流氓波是线流氓波，它产生于恒定的背景线轮廓，然后消失在恒定的背景。它还表明，多流氓波描述的几个基本流氓波的相互作用。这些多流氓波也产生于恒定的背景，然后衰减回它，但在中间时间，有趣的弯曲波图案出现。然而，高阶流氓波被发现显示出更有趣的功能。具体地说，只有部分的波结构在高阶流氓波从恒定的背景上升，然后退回到它，这种瞬态波表现出新的模式，如抛物线。但波结构的另一部分来自远距离，作为局部块，它减速到近场并与瞬态流氓波相互作用，然后被反射回来并再次加速到大距离。这些流氓波的解决方案有有趣的影响，在海洋中的二维表面水波。
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。
Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations
针对Kadomtsev-Petviashvili和Davey-Stewartson方程，比较了多种四阶时间步进法（指数时间差分、积分因子、时间分裂、隐式Runge-Kutta和Driscoll复合Runge-Kutta）在数值守恒律方面的性能。
Semi-rational solutions of the third-type Davey-Stewartson equation
利用双线性方法导出了第三类Davey-Stewartson（DS-III）方程的一般暗孤子和由暗孤子和呼吸子组成的混合解。通过引入这两个微分算子，得到了由流氓波、呼吸子和孤子组成的半有理解。这些半有理解是以行列式的形式给出的，其矩阵元素具有简单的代数表达式。在适当的参数条件下，我们得到了一般的流氓波的解表示的有理函数。它表明，基本（最简单的）流氓波是线流氓波。它还表明，多流氓波描述了几个基本流氓波的相互作用，这将产生有趣的弯曲波图案。高阶流氓波起源于一个局部块体，然后退回到块体中，并给出了几种由流氓波、呼吸子和孤子组成的混合解。具体地说，这些半有理解有一个新的现象：在暗孤子上形成团块，并观察到从暗孤子逐渐分离。
Variable coefficient Davey-Stewartson system with a Kac-Moody-Virasoro symmetry algebra
研究了变系数Davey-Stewartson（vcDS）系统的对称群性质。证明了具有Kac-Moody-Virasoro（KMV）结构的李点对称代数与通常的（常系数）DS系统同构的充要条件是系数满足一定的条件.这些条件与vcDS系统通过点变换可变换为DS系统的条件一致。利用vcDS系统的等价群从一类不可积子系统中选出可积子系统。此外，完整的对称群的DS系统推导明确没有指数的对称代数。Ozawa为DS系统引入的集块解（在$\mathbb{R}^2$中在各个方向上合理地局部化），在系统属于可积类，即与DS系统等价时，甚至对vcDS系统也成立.这些解可用于建立聚焦情形下空间$L^2(\mathbb{R}^2)$中有限时间的精确爆破解。
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。
Dynamics of Rogue Waves in the Partially PT-symmetric Nonlocal Davey-Stewartson Systems
本文研究了部分$\cal{PT}$对称非局域Davey-Stewartson（DS）系统中的流氓波动力学.利用达布变换方法，导出了部分$\cal{PT}$对称非局部DS方程中的一般异常波.对于部分$\cal{PT}$ -对称非局部DS-I方程，得到了其解，并用行列式表示.对于部分$\cal{PT}$对称的DS-II方程，其解表示为拟Gram行列式.结果表明，这两个系统中的基本异常波都是有理解，它产生于$t\rightarrow -\infty$处的恒定背景，并在临界时刻在空间平面上的整个双曲线上发展出有限时间奇异性。文中还指出，几个基本无赖波的相互作用可以用多无赖波来描述。基反常波与暗和反暗理性行波相互作用产生了新的混合波型。然而，没有高阶流氓波被发现在这个部分$\cal{PT}$对称的非局部DS系统。相反，它可以从高阶有理解产生一些高阶行波。
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…
Spatially-bounded rogue waves in the Davey-Stewartson I equation
本文研究了Davey-Stewartson I方程中的空间受限rogue波，通过高阶有理解和非退化条件推导出其形成机制及渐近近似。
In this work we present spectral algorithms for the numerical scattering for the defocusing Davey-Stewartson (DS) II equation with initial data having compact support on a disk, i.e., for the solution of d-bar problems. Our algorithms use polar coordinates and implement a Chebychev spectral scheme for the radial dependence and a Fourier spectral method for the azimuthal dependence. The focus is placed on the construction of complex geometric optics (CGO) solutions which are needed in the scattering approach for DS. We discuss two different approaches: The first constructs a fundamental solutio…
We present an efficient high-precision numerical approach for the Davey-Stewartson (DS) II equation, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll's composite Runge-Kutta method for the time dependence. Since the DS equation is a nonlocal, nonlinear Schrödinger equation with a singular symbol for the nonlocality, standard Fourier methods in practice only reach accuracies of the order of $10^{-6}$ or less for typical examples. This was…
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。
A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation in the Semiclassical Limit
在数值实验中，散焦的戴维-斯图尔特森II方程在半经典极限中表现出的行为在性质上类似于其一维约化的散焦非线性薛定谔方程，即从光滑的初始数据产生规则的快速振荡，占据在极限中变得明确定义的时空域。作为第一步，研究这个问题的分析使用逆散射变换，我们考虑直接谱变换的散焦Davey-Stewartson II方程光滑的初始数据在半经典极限。直接谱变换涉及二维奇摄动椭圆Dirac系统。我们引入了一个WKB型的方法来解决这个问题，证明了它是有意义的正式足够大的值的谱参数$k$通过控制相关的非线性程函问题的解决方案，我们给出的数值证据表明，该方法是准确的，这样的$k$在半经典极限。产生这个证据需要奇摄动狄拉克系统的数值解和程函问题的数值解。前者是使用一种方法以前开发的两个作者，我们在本文中给出了一个新的方法，数值解的程函问题有效的足够大的$k$。对于一个特定的潜在的，我们能够解决程函问题的封闭形式的所有$k$，计算，产生一些洞察失败的WKB方法的$k$值较小。通知的直接谱变换的数值计算，然后我们开始研究奇摄动狄拉克系统的值$k$如此之小，没有全球解决方案的程函问题。
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。
Whitham modulation theory for the Davey-Stewartson system and stability analysis of its periodic traveling wave solutions
本文为Davey-Stewartson系统建立Whitham调制理论，分析其四种变体周期行波解的稳定性，发现仅散焦DSII系统线性稳定，其余均不稳定，并通过数值模拟验证了理论。
Discrete symmetry's chains and links between integrable equations
研究了非线性薛定谔方程（NLS）和Davey-Stewartson方程（DS）的离散对称性缀饰链。得到了修正的NLS方程和修正的DS方程。构造了mNLS和mDS方程的显可逆Backlund自变换。我们证明了KP和DS模型的离散对称性共轭链。得到了P4方程的二维推广。
Quasiclassical generalized Weierstrass representation and dispersionless DS equation
提出了四维欧氏空间中慢调制高度波纹曲面的准经典广义Weierstrass表示。这种曲面的可积变形由无色散的Davey-Stewartson族描述。本文还讨论了其它四维空间和无色散DS系统的准经典GWR。
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