Alexey Samokhin
An arbitrary compact-support initial datum for the Korteweg-de Vries equation asymptotically splits into solitons and a radiation tail, moving in opposite direction. We give asimple method to predict the number and amplitudes of resulting solitons and some integral characteristics of the tail using only conservation laws.
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Infinitely many conservation laws for the discrete KdV equation
在\cite{RH3}拉辛和海顿提出了一种方法来构建一个无限数量的守恒定律的离散KdV方程（dKdV），通过重复应用一定的对称性，以一个已知的守恒定律。然而，没有决定由此产生的守恒定律是否是独特的和非平凡的。本文得到了如下结果：（1）给出了用离散的Gardner变换构造无穷多个守恒律的另一种方法。(2)我们给出了一个直接的证明，拉辛-海顿守恒律确实是独特的和非平凡的。(3)我们考虑一个连续极限，其中dKdV方程变为一阶程函方程。在这个极限中，两组守恒律变得相同，并且显然是不同的和非平凡的。这证明了Gardner方法所构造的守恒律的非平凡性，并给出了Rasin-Hydon方法所构造的守恒律的非平凡性的另一种证明.
The degenerate coupled multi-KdV equations for coupled multiplicity l=3 are studied. The equations also known as three fields Kaup-Boussinesq equations are considered for invariant analysis and conservation laws. The classical Lie's symmetry method is used to analyze the symmetries of equations. Based on Killing's form which is invariant of adjoint action, the full classification for Lie algebra is presented. Further, one-dimensional optimal group classification is used to obtain invariant solutions. Besides this, using general theorem proved by Ibragimov we find several nonlocal conservation …
Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion
本文对广义拟线性KdV方程进行了点对称性和低阶局部守恒律的完整分类，给出了对称变换群的解释、守恒量的物理解释，并获得了少数行波解。
Deformations of dispersionless KdV hierarchies
研究了多组分无色散KdV系统存在流体力学守恒律族的障碍。它表明，如果一个潜在的代数是约旦，那么最低的障碍消失，所有更高的障碍自动消失。这些多组分的色散KdV型方程的变形进行了研究。没有新的障碍物出现，因此，完全变形的层次的存在取决于一个单一的纯流体力学守恒定律的存在。
Conservation laws for a generalized seventh order KdV equation
本文用乘子法完整分类了广义七阶KdV方程的低阶局部守恒律，用李方法分类了点对称性，发现无额外对称性，并确定了约化方程的积分因子。
Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the u…
Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
对于一类与仿射（扭曲或非扭曲）卡茨-穆迪代数相关的广义可积族，利用单个标量τ-函数导出了其局部守恒密度的显式表示.该τ函数作为守恒密度的配分函数，这符合其作为某些量子系统的有效作用的潜在解释。这个类由Drinfel 'd-Sokolov和二维仿射Toda晶格层次的多分量推广组成。本文还讨论了前者与Feigin，Frenkel和Enriquez对KdV和mKdV型孤子方程的研究方法之间的关系。这些结果大大简化了计算所携带的孤子解决方案的方程的层次，这是重要的，以建立他们的解释为粒子的保守电荷。作为例子，我们计算了最近构造的一组约束KP孤子所携带的电荷。
Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this letter we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation. The derivation is based on the fact that the scattering data $a(z)$ is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schrödinger spectral problem. The obtained conserved densities are different from those in the known literatures. They are asymptotic to z…
Coupled system description of perturbed KdV equation
在多孤子情形下，利用扰动KdV方程解展开的自由度，将方程转化为两个方程的系统：KdV型孤子的（可积）标准形，它服从通常的无穷大KdV守恒律;和一个辅助方程，它描述了障碍物对渐近可积性的贡献，它产生于二阶以上。在展开式中，分析一直进行到三阶。在该阶内，辅助方程的解是守恒量。
It is shown that equations of the Korteweg-de Vries hierarchy and their conservation laws can be expressed via the whole powers of an integro-differential operator and functions provided by them.
Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies
本文在Sato理论的框架下讨论了非对易空间上Lax族的交换流和守恒律。在交换空间上，Sato理论揭示了由Lax族导出的一类孤子方程在伪微分算子意义下的可积性。Sato理论的非交换扩张已经被作者和户田光一研究过，并且保证了各种非交换Lax族的存在性。本文给出了同时具有空-空和空-时非对易性的非对易Lax族的守恒律，证明了无穷多个守恒密度的存在性.我们还用Lax算子给出了它们的显式表示。我们的结果包括KP、KdV、Boussinesq、耦合KdV、Sawada-Kotera、修正KdV方程等非对易形式。
Soliton Shielding of the focusing modified KdV equation
本文研究聚焦mKdV方程孤子气体在点谱位于有界区域时的确定性行为，提出并证明了求积域和椭圆域条件下的孤子屏蔽现象及谱数据约化结果。
Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations
给出了求解非线性偏微分方程和微分差分方程守恒律的直接方法和符号软件，并将其应用于（1 + 1）维非线性偏微分方程，包括Korteweg-de Vries（KdV）方程、Boussinesq方程和Drinfel 'd-Sokolov-Wilson方程。将这些方法的改编应用于具有超越非线性的偏微分方程。例子包括sine-Gordon、sinh-Gordon和Liouville方程。相对于非线性DDE，我们的方法被应用到卡茨-范Moerbeke，户田，和Ablowitz-拉迪克格。为了克服待定系数法的不足，我们设计了一种新的直接法，它采用了超前阶分析。该方法适用于离散化的KdV和修改的KdV方程，及其组合。其他例子包括由于Bogoyavlenskii，Belov-Chaltikian和Blaszak-Marciniak的晶格。在Mathematica中实现了偏微分方程和微分方程的待定系数法。代码"transPDEDensityFlux. m"计算具有或不具有超越非线性的偏微分方程系统的密度和通量。代码"DDEDensityFlux. m"对多项式非线性DDE执行相同的操作。从首阶项开始，新的Maple库"离散"计算非线性DDE的密度和通量。
Asymptotically conserved charges and 2-kink collision in quasi-integrable potential KdV models
本文研究势 KdV 模型的一种特定形变，构造准守恒律并数值验证双扭结弹性散射，同时提出反常抵消机制与重整化程序以处理发散荷。
(Non)local Hamiltonian and symplectic structures, recursions, and hierarchies: a new approach and applications to the N=1 supersymmetric KdV equation
利用数学方法DG/0304245和[I. S. Krasil'shchik和P. H. M. Kersten，古典和超对称微分方程的对称性和递归算子，Kluwer，2000]，我们完成了N=1超对称Korteweg-de Vries方程的广泛研究。结果包括：描述当地和非当地的哈密顿和辛结构，五个层次的对称性，相应的层次的守恒定律，递归算子的对称性和生成函数的守恒定律。我们强调，本文的重点不仅仅是对super-KdV方程本身的结果，而仅仅是对几何方法和基于它的计算算法的有效性的阐述。
We found, through analytical and numerical methods, new towers of infinite number of asymptotically conserved charges for deformations of the Korteweg-de Vries equation (KdV). It is shown analytically that the standard KdV also exhibits some towers of infinite number of anomalous charges, and that their relevant anomalies vanish for $N-$soliton solution. Some deformations of the KdV model are performed through the Riccati-type pseudo-potential approach, and infinite number of exact non-local conservation laws is provided using a linear formulation of the deformed model. In order to check the d…
We consider the construction of the deformed Whitham system for the KdV-equation in the one-phase case and investigate the conservation of the Hamiltonian properties in this situation. It is shown then, that both the Gardner - Zakharov - Faddeev and the Magri brackets give the deformed Dubrovin - Novikov brackets (the brackets of Dubrovin - Zhang type) for the deformed Whitham system constructed by our procedure. The general approach used in the paper gives a scheme for the averaging of the Poisson structures in the general situation.
In this paper, based on the regular KdV system, we study negative order KdV (NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many conservation laws, and explicit multi-soliton and multi-kink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative order KdV hierarchy. The NKdV equations are not only gauge-equivalent to the Camassa-Holm equation through some hodograph transformations, but also closely related to the Ermakov-Pinney systems, and the Kuper…
The Korteweg-de Vries limit for the global dynamics of the Toda lattice
本文研究Toda晶格在连续极限下的动力学行为，证明其H^1初始数据解在KdV缩放和适当平移下全局收敛于KdV方程解，利用调和分析工具和Toda晶格的守恒律。
Riccati-type pseudo-potential approach to quasi-integrability of deformed soliton theories
本文综述了Riccati型伪势方法在准可积sine-Gordon、KdV和NLS模型中的应用，提出统一框架分析多种变形可积系统，发现异常守恒定律和非局域守恒定律，对非线性物理多个领域具有潜在应用价值。
Nonlinear generalizations of integrable equations in one dimension, such as the KdV and Boussinesq equations with $p$-power nonlinearities, arise in many physical applications and are interesting in analysis due to critical behaviour. This paper studies analogous nonlinear $p$-power generalizations of the integrable KP equation and the Boussinesq equation in two dimensions. Several results are obtained. First, for all $p\neq 0$, a Hamiltonian formulation of both generalized equations is given. Second, all Lie symmetries are derived, including any that exist for special powers $p\neq0$. Third, …
We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found to two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair we construct a Miura-type transformation relating the original equation to the KdV equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a …
Evolution of internal cnoidal waves with local defects in a two-layer fluid with rotation
研究了具有旋转的两层流体中内部波的演化，提出了一种渐近方法，能够获得Ostrovsky方程的解，并展示了局部缺陷初始条件可导致大内部波和剪切电流的爆发。
The Coupled Modified Korteweg-de Vries Equations
研究了修正的KdV方程在多组分体系中的推广，即用$(\partial u_i)/(\partial t) + 6 (\sum_{j,k=0}^{M-1} C_{jk} u_j u_k) (\partial u_i)/(\partial x) + (\partial^3 u_{i})/(\partial x^3) = 0, i=0, 1, ..., M-1 $表示。我们应用一个新的扩展版本的逆散射方法，这个系统。结果表明，该系统具有无穷多个守恒律和多孤子解。进一步解决了模型的初值问题。
Inelastic Scattering, Emergent Interactions of Solitons in the Zakharov-Kuznetsov Equation through Conservative and non-Conservative Physics-Informed Neural Networks
本文利用物理信息神经网络研究2+1维Zakharov-Kuznetsov方程的孤子碰撞，通过引入保守PINNs解决了碰撞过程不遵循守恒定律的问题，并揭示了碰撞中系数变化引起的临时相互作用涌现。
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