Asymptotics of the solitary waves for the generalised Kadomtsev-Petviashvili equations
Philippe Gravejat
本文研究了广义Kadomtsev-Petviashvili方程局部孤波的渐近行为。特别地，我们计算了它们在任意维数$N \geq 2$上的一阶渐近性。
We investigate the asymptotic behaviour of the localised solitary waves for the generalised Kadomtsev-Petviashvili equations. In particular, we compute their first order asymptotics in any dimension $N \geq 2$.
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In this paper, we study the existence and concentration of solitary waves for a class of generalized Kadomtsev-Petviashvili equations with the potential in $\mathbb{R}^2$ via the variational methods.
We show that the fifth-order Kadomtsev-Petviashvili II equation is globally well-posed in an anisotropic Gevrey space, which complements earlier results on the well-posedness of this equation in anisotropic Sobolev spaces.
The Cauchy problem for the fractional Kadomtsev-Petviashvili equations
本文的目的是证明一类分数阶Kadomtsev-Petviashvili（KP）方程的Cauchy问题的各种不适定性和适定性结果，包括Benjamin-Ono方程的KP形式和中间长波方程.
In this paper we use variational methods to establish a Berestycki-Lions type result for a class of generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^N$. The positive and zero mass cases are considered.
Results of stabilization for the higher order of the Kadomtsev-Petviashvili equation are presented in this manuscript. Precisely, we prove with two different approaches that under the presence of a damping mechanism and an internal delay term (anti-damping) the solutions of the Kawahara-Kadomtsev-Petviashvili equation are locally and globally exponentially stable. The main novelty is that we present the optimal constant, as well as the minimal time, that ensures that the energy associated with this system goes to zero exponentially.
On the Cauchy-problem for generalized Kadomtsev-Petviashvili-II equations
证明了广义Kadomtsev-Petviashvili-II方程$$u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3,$$的Cauchy问题在几乎临界各向异性索伯列夫空间中的局部适定性.在$X_{s,b}$ -空间中，通过多重线性插值，证明了局部光滑和极大函数估计以及双线性加细的Lithartz型不等式.
Soliton asymptotics of rear part of non-localized solutions of the Kadomtsev-Petviashvili equation
我们构造了Kadomtsev-Petviashvili-I方程的非局部化的，真正的整体解，这些解对于$x\to-\infty$是消失的，并研究了它们的大时间渐近行为。我们证明，这样的解决方案（$t\to\infty$）推出一个训练的弯曲渐近孤子的基本波包后面移动。
Existence, regularity and concentration phenomenon of nontrivial solitary waves for a class of Generalized Kadomtsev-Petviashvili (GKP) equation in $\mathbb{R}^2$
本文在$\mathbb{R}^2.$中建立了一类广义Kadomtsev-Petviashvili（GKP）方程非平凡孤波的存在性、正则性和集中现象的一些结果。变分方法被用来得到一个存在性结果并研究集中现象，而正则性由于我们是在各向异性的索伯列夫空间中引入函数而更加微妙。
Low regularity solutions for the Kadomtsev-Petviashvili I equation
本文在能量空间和自然加权$L^{2}$空间的交空间中，得到了小数据量下KP-I方程的局部时间存在性、（适当的）唯一性和连续依赖性.
Well-posedness of the Fifth Order Kadomtsev-Petviashvili I Equation in Anisotropic Sobolev Spaces with Nonnegative Indices
本文在具有非负指标的各向异性索伯列夫空间中建立了实值五阶Kadomstev-Petviashvili I方程的局部和整体适定性.特别地，我们的局部适定性改进了Saut-Tzvetkov的适定性，而我们的全局适定性则对Saut-Tzvetkov的$L^2$ -数据猜想给出了肯定的回答.
Of concern are traveling wave solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. The existence of periodically modulated solitary wave solutions is proved by dimension breaking bifurcation. Moreover, the line solitary wave solutions and their transverse (in)stability are discussed. Analogous to the classical Kadmomtsev--Petviashvili (KP) equation, the fKP equation comes in two versions: fKP-I and fKP-II. We show that the line solitary waves of fKP-I equation are transversely linearly instable. We also perform numerical experiments to observe the (in)stability dynamics of line…
On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation
通过在Bourgain型空间中的工作，证明了Kadomotsev-Petviashvili-Burgers-I方程（KPBI）的Cauchy问题在$H^{s,0}(\mathbb{R}^{2})$，$s > -1/2$中的局部和全局适定性.如果要求流图平滑，这个结果几乎是尖锐的。
On solutions for the Kadomtsev-Petviashvili I equation
利用振荡积分技术研究了KP-I方程的适定性，其中初始数据相对于加权索伯列夫空间的范数较小，且其导数的总阶数不大于2.
Stability of the line soliton of the Kadomtsev--Petviashvili-I equation with the critical traveling speed
本文研究了Kadomtsev-Petviashvili-I方程在$\mathbb{R} \times (\mathbb{R}/2π\mathbb{Z})$中孤子的轨道稳定性，$\mathbb{R} \times (\mathbb{R}/2π\mathbb{Z})$是Korteweg-de Vries方程的高维推广. Benjamin证明了Korteweg-de Vries方程存在稳定的单孤子。我们把Korteweg-de Vries方程的单孤子看作Kadomtsev-Petviashvili-I方程的线孤子。Zakharov和Zakset-Tzvetkov证明了$\mathbb{R}^2$上Kadomtsev-Petviashvili-I方程的线孤子的轨道不稳定性。对于$\mathbb{R} \times (\mathbb{R}/2π\mathbb{Z})$上的Kadomtsev-Petviashvili-I方程，利用Tzvetkov集证明了线孤子随传播速度$c>4/\sqrt{3}$的轨道不稳定性和线孤子随传播速度$0<c<4/\sqrt{3}$的轨道稳定性.本文证明了Kadomtsev-Petviashvili-I方程在$\mathbb{R} \times (\mathbb{R}/2π\mathbb{Z})$上的线孤子在临界速度$c=4/\sqrt{3}$下的轨道稳定性以及线孤…
Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations
在非各向同性索伯列夫空间H^{s_1，s_2}（R^2）中，当s_1 > -1/2和s_2 \geq 0时，我们证明了Kadomtsev-Petviashvili II方程初值问题在时间上的局部适定性.在H^{s_1，0}（R^2）标度上，该结果包括整个亚临界范围，而无需对初始数据进行任何额外的低频假设。更一般地，我们证明了当地的时间适定性的柯西问题的色散广义KP II型方程。我们还推导出一个整体的适定性结果的广义方程。
Transverse spectral Instabilities in rotation-modified Kadomtsev-Petviashvili equation and related models
该研究针对旋转修正Kadomtsev-Petviashvili方程及相关模型，分析Ostrovsky方程小周期行波的横向稳定性，还推广到含一般色散、二次三次非线性的广义模型，得到多种相关二维方程的横向稳定/不稳定性结论。
Regular Kadomtsev-Petviashvili (KP) solitons have been investigated and classified successfully by the Grassmannian. We provide rigorous analysis for the direct scattering problem of perturbed $\textrm{Gr}(1, 2)_{\ge 0}$ KP solitons.
Large-Time Asymptotics for the Kadomtsev-Petviashvili I Equation
本文利用逆散射方法，在排除 lump 解的初值假设下，证明了 KP I 方程小初值解的大时间渐近性，并精确描述了辐射场。
Stability of Kadomtsev-Petviashvili multi-line solitons
该研究首次严格证明了含连续与离散散射数据且连续散射数据支集非退化的多维可积系统的逆散射理论，进而论证了Kadomtsev-Petviashvili多线孤子的L^∞稳定性。
Regular Kadomtsev-Petviashvili II (KPII) line solitons have been investigated and classified successfully by the Grassmannians. The inverse scattering method provides a promising and powerful approach to study the stability properties of $\textrm{Gr}(N, M)_{> 0}$ KP solitons. In this paper, we complete rigorous analysis for the direct scattering problem of perturbed $\textrm{Gr}(N, M)_{> 0}$ KP solitons}.
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 …
The Kadomtsev-Petviashvili equation in conformal variables for waves over topography
本文通过保形映射方法推导了考虑地形影响的Kadomtsev-Petviashvili方程，该方程适用于弱非线性表面波，且地形无需为光滑函数。
We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation \begin{align*} \partial_{x}\left(u_{t}-β\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-γu=0 \end{align*} in the anisotropic Sobolev spaces $H^{s_{1},\>s_{2}}(\mathbb{R}^{2})$. When $β<0$ and $γ>0,$ we prove that the Cauchy problem is locally well-posed in $H^{s_{1},\>s_{2}}(\mathbb{R}^{2})$ with $s_{1}>-\frac{1}{2}$ and $s_{2}\geq 0$. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang( Transactions of the American Mathematical Society, 364(2012), 339…
Asymptotic profile of solutions to the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with anisotropic dissipation in 2D
本文研究了二维广义Kadomtsev-Petviashvili方程带各向异性耗散的Cauchy问题的渐进行为，证明了解的$L^{\infty}$-范数衰减率及渐近剖面。
Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations
本文首先回顾了关于KP型方程的已知数学结果。然后，我们进行数值模拟分析各种定性性质的方程：爆破与长时间的行为，稳定性和不稳定的孤波。
Lump solutions of the fractional Kadomtsev--Petviashvili equation
本文用变分方法证明能量次临界 α>4/5 时 fKP-I 方程存在非平凡光滑 lump 解且二次衰减，并通过数值实验验证存在性、衰减及截面对称性。
The inverse scattering theory of Kadomtsev-Petviashvili II equations
该研究概述KP II方程的逆散射理论，聚焦受扰多线孤子逆问题，证明不同背景下的逆散射理论具有一致性。
Blow-Up Theory and Liouville-Type Theorem for Solutions of a Class of Generalized Camassa-Holm-Kadomtsev-Petviashvili Equations
本文研究了一类广义CH-KP方程解的爆发行为和Liouville型定理，通过特征线、先验估计和Riccati不等式证明了爆发准则和唯一性定理。
Transverse stability of periodic traveling waves in Kadomtsev-Petviashvili equations: A numerical study
我们数值研究了所谓的椭圆余弦波的横向稳定性和不稳定性，Kadomtsev-Petviashvili方程的时间演化下Korteweg-de Vries方程的周期行波解。特别是，我们发现，在KP-I小振幅椭圆余弦波是稳定的（至少对于空间局部扰动），只有在一定的阈值以上才变得不稳定。与此相反，KP-II被发现是稳定的所有振幅，或等效地，波速。这与最近在\cite{RT1, RT2}中给出的孤波分析结果是一致的。
We prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev-Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev-Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev-Petviashvili is locally well-posed in $H^s(\mathbb R^2)$, for $s>\frac74$, in the capillary-gravity setting.
On a rod-Kadomtsev-Petviashvili shallow water equation in two dimensions
本文从三维 Euler 方程导出二维 rod-KP 方程，证明局部适定性，通过守恒律构造全局解或有限时间爆破解，并研究行波解与对称性。
The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$
本文通过新的$L^4$-Strichartz估计和双线性Strichartz估计，解决了周期性KP-II方程在负Sobolev正则性初始数据下的Cauchy问题。
Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations
研究广义Kadomtsev-Petviashvili方程在临界和超临界非线性下的解行为，通过动态缩放识别奇点类型，讨论观测到的爆破场景。
On critical behaviour in generalized Kadomtsev--Petviashvili equations
本文给出了广义Kadomtsev-Petviashvili（KP）方程解中色散激波形成的渐近描述。基于多尺度展开给出了Painleve' I族常微分方程特解的渐近描述。几个例子进行了数值讨论，提供有力的证据的有效性的猜想。
Large time behavior and optimal decay estimate for solutions to the generalized Kadomtsev--Petviashvili--Burgers equation in 2D
本文研究二维广义Kadomtsev–Petviashvili–Burgers方程Cauchy问题，构造解在大时间下的近似公式，并通过$L^{\infty}$范数下界证明已知的$t^{-\frac{7}{4}}$衰减速率是最优的。
Sharp well-posedness for Kadomtsev-Petviashvili-Burgers (KPBII) equation in $R^2$
本文证明了Kadomotsev-Petviashvili-Burgers方程（KPBII）的Cauchy问题在$\mathbb R^2$中的整体适定性，当初值属于各向异性索伯列夫空间$H^{s_1,s_2}(\mathbb R^2)$时，对所有的$s_1>-\frac12$和$s_2\geq0$都是如此.另一方面，我们在某种意义上证明了我们的结果是尖锐的。
Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation
本文研究了线性系统与Kadomtsev-Petviashvili方程的解，通过Fredholm行列式和Hankel算子的性质，建立了与Pöppe半加性算子相关的数学框架，为数值求解KP方程提供了有效方法。
We study transverse stability and instability of one-dimensional small-amplitude periodic traveling waves of a generalized Kadomtsev-Petviashvili equation with respect to two-dimensional perturbations, which are either periodic or square-integrable in the direction of the propagation of the underlying one-dimensional wave and periodic in the transverse direction. We obtain transverse instability results in KP-fKdV, KP-ILW, and KP-Whitham equations. Moreover, assuming the spectral stability of one-dimensional wave with respect to one-dimensional square-integrable periodic perturbations, we obta…
We classify the Lie point symmetries for the 2+1 nonlinear generalized Kadomtsev-Petviashvili equation by determine all the possible f(u) functional forms where the latter depends. For each case the one-dimensional optimal system is derived; a necessary analysis to find all the possible similarity transformations which simplify the equation. We demonstrate our results by constructing static and travel-wave similarity solutions. In particular the latter solutions satisfy a second-order nonlinear ordinary differential equation which can be solved by quadratures.
Critical lengths for the linear Kadomtsev-Petviashvili II equation
本文研究了二维卡多莫特塞夫-彼得萨夫斯基 II 方程的临界长度问题，通过推导可观察性不等式，实现了线性系统的精确边界可控性和边界指数稳定化。
On the Gevrey regularity of the fifth-order Kadomtsev-Petviashvili-II equation: An improved approach
本文改进分析方法，扩展五阶Kadomtsev-Petviashvili-II方程解的Gevrey正则性结果，证明2+1维解时间正则阶为5σ（初始空间正则阶σ≥1），且给出色散方程时间Gevrey正则的一般原理。
The Cauchy problem for two dimensional generalized Kadomtsev-Petviashvili-I equation in anisotropic Sobolev spaces
本文的目标有三个方面。首先证明了广义KP-I方程\begin{eqnarray*} u_{t}+|D_{x}|^α\partial_{x}u+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,α\geq4 \end{eqnarray*}的Cauchy问题在各向异性索伯列夫空间$ H^{s_{1},\>s_{2}}(\R^{2})$中的局部适定性，其中有$s_{1}>-\frac{α-1}{4}$和$s_{2}\geq 0$ .其次，我们证明了问题在具有$s_{1}>-\frac{(α-1)(3α-4)}{4(5α+3)}$的$H^{s_{1},\>0}(\R^{2})$中是全局适定的，如果$4\leq α\leq5$。最后，证明了当满足$α>5$时，该问题在具有$s_{1}>-\frac{α(3α-4)}{4(5α+4)}$的$H^{s_{1},\>0}(\R^{2})$中是全局适定的.我们的结果改进了Saut和Tzvetkov（J. Math. Pures Appl.79（2000），307-338）的结果。以及Li和Xiao（J.Math.PuresAppl.90（2008），338-352.）。
We investigate the Cauchy problem on the cylinder, namely the semi-periodic problem where there is periodicity in the $x$-direction and decay in the $y$-direction, for the Kadomtsev-Petviashvili II equation by the inverse spectral transform method. For initial data with small $L^1$ and $L^2$ norms, assuming the zero mass constraint, this initial-value problem is reduced to a Riemann-Hilbert problem on the boundary of certain infinite strips with shift.
Singularity analysis of a spherical Kadomtsev-Petviashvili equation
本文研究了J. - K.薛[Phys. Lett. A 314：479-483（2003）]在最高共振处的可积性的Painleve测试失败，其中出现递归关系的非平凡相容性条件。然而，这个相容性条件是足够弱的，因此允许SKP方程具有可积的（1+1）维约化，这是由截断奇异展开的Weiss方法检测到的。
Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation
本文研究了广义Kadomtsev-Petviashvili方程的周期行波解对长波长横向扰动的谱不稳定性.通过分析适当的周期性埃文斯函数的高，低频率限制，我们推导出一个取向指数，产生充分条件，这样的不稳定性发生。这个指数是几何性质的，适用于任意周期性行波与小光滑和凸性假设的非线性。利用控制行波剖面的常微分方程的可积结构，我们能够计算Korteweg-de Vries方程和修正Korteweg-de Vries方程的椭圆函数解的取向指数。
We consider the Kadomtsev-Petviashvili (KP) equations posed on $\mathbb{R}^2$. For both equations, we provide sequential in time asymptotic descriptions of solutions, of arbitrarily large data, inside regions not containing lumps or line solitons, and under minimal regularity assumptions. The proof involves the introduction of two new virial identities adapted to the KP dynamics, showing decay in large regions of space, especially in the KP-I case, where no monotonicity property was previously known. Our results do not require the use of the integrability of KP and are adaptable to well-posed …
The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI
建立了一类特殊的Painleve VI方程之间的等价性，参数化的共形维数$μ$，时间相关的Euler top方程，isomonodromic变形和三维Frobenius流形。通过Grassmannian公式，利用2-向量1-约束辛Kadomtsev-Petviashvili（CKP）族的Wronskian解，构造了Euler顶方程的等序τ函数和解.这些朗斯基解给出了Painleve VI方程$μ=1,2,{...} $的有理解。
The Kadomtsev-Petviashvili I Equation on the Half-Plane
\cite{F1997}于1997年提出了一种求解一维\textit{linear}和\textit{integrable nonlinear}演化偏微分方程初边值问题的新方法.这种方法随后被扩展到两个空间维度的演化偏微分方程的初边值问题，首先是线性偏微分方程\cite{F2002b}的情况，最近，在可积非线性偏微分方程的情况下，半平面上的Davey-Stewartson和Kadomtsev-Petviashvili II方程（分别参见\cite{FDS2009}和\cite{MF2011}）。本文研究了Kadomtsev-Petviashvili I方程的类似问题;特别地，通过经由d-杆形式主义的相关Lax对的同时谱分析，我们能够获得解的整表示，其涉及所有初始值和边界值的某些变换，以及恒等式，所谓的整体关系，其在复谱平面的适当区域中关联这些变换。
We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator for our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral oper…
Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev--Petviashvili equation
KP-I方程\[ (u_t-2uu_x+\tfrac{1}{2}(β-\tfrac{1}{3})u_{xxx})_x -u_{yy}=0 \]作为具有强表面张力（Bond数$β>1/3$）的重力-毛细波的弱非线性模型方程而出现。这个方程承认-作为一个明确的解决方案-一个“完全本地化”或“块”孤立波衰减到零在所有空间方向。最近，人们对\emph{full-dispersion KP-I equation} \[u_t + m({\mathrm D}) u_x + 2 u u_x = 0,\]产生了兴趣，其中$m({\mathrm D})$是具有符号\[ m(k) = \left( 1 + β|k|^2|\right)^{\frac{1}{2}} \left( \frac{\tanh |k|}{|k|} \right)^{\frac{1}{2}} \left( 1 + \frac{2k_2^2}{k_1^2} \right)^{\frac{1}{2}}, \]的傅立叶乘法器，其通过保留来自水波问题的精确色散关系而获得。在本文中，我们表明，FDKP-I方程也有一个完全本地化的孤波解。存在理论本质上是变分的和微扰的。完全局部化孤波的变分原理被简化为一个局部等价的变分原理，其特征在于与{KP-I}方程的完全局部化孤波解相关的变分泛函的扰动。通过在其自然约束集上最小化该约化泛函，找到了该约化泛…
On the low regularity of the fifth order Kadomtsev-Petviashvili I equation
我们考虑五阶Kadomtsev-Petviashvili I（KP-I）方程为$\partial_tu+α\partial_x^3u+\partial^5_xu+\partial_x^{-1}\partial_y^2u+uu_x=0,$，而$α\in \mathbb{R}$ .引入一个插值能量空间$E_s$来研究五阶KP-I方程的初值问题的适定性.我们得到了五阶KP-I方程在$0<s\leq1$中IVP的局部适定性。为了获得局部适定性，我们提出了一个双线性估计的Bourgain空间的框架内的能量空间。它主要取决于并元分解的Lahartz估计、五阶色散平滑效应和最大估计。关键词：五阶KP-I方程，Bourgain空间，并矢分解的Schmidhartz估计，色散平滑效应，极大估计。
Existence and multiplicity of normalized solutions for the generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^2$
本文首次在$L^2$-约束下研究广义Kadomtsev-Petviashvili方程的规范化解，针对次临界和超临界情形建立了基态解的存在性，并在组合非线性情形下证明了局部极小基态解及正能量第二解的多重性。
In this paper, we study the asymptotic behavior of weakly transverse water-waves under a weak Coriolis forcing in the long wave regime. We derive the Boussinesq-Coriolis equations in this setting and we provide a rigorous justification of this model. Then, from these equations, we derive two other asymptotic models. When the Coriolis forcing is weak, we fully justify the rotation-modified Kadomtsev-Petviashvili equation (also called Grimshaw-Melville equation). When the Coriolis forcing is very weak, we rigorously justify the Kadomtsev-Petviashvili equation. This work provides the first mathem…
Numerical solution of the generalized Kadomtsev-Petviashvili equations with compact finite difference schemes
本文提出紧致有限差分方案求解广义Kadomtsev-Petviashvili方程，对比傅里叶谱方法，验证了数值收敛性，并研究了孤子不稳定性及爆破行为。
The Kadomtsev-Petviashvili II Equation on the Half-Plane
KPII方程是2+1维（两个空间和一个时间）中的可积非线性PDE，它出现在几种物理环境中，包括流体力学，其中它描述了浅水中的波。它提供了一个多维的概括著名的KdV方程。在这项工作中，我们采用了一种新的方法，最近推出的作者之一，在连接的Davey-Stewartson方程\cite{FDS2009}，为了分析的初边值问题的KPII方程制定的半平面。该分析关键使用所谓的d-杆形式主义，以及所谓的全球关系。与2+1中的初值问题相反，边界的一个新特征是d-杆形式主义现在涉及复平面中的一个函数，该函数在实轴上是不连续的。
Lie symmetry analysis of (2+1)-dimensional time fractional Kadomtsev-Petviashvili equation
本文利用Lie对称分析研究(2+1)维时间分数阶KP方程，通过对称约化得到低维分数阶方程及其精确解，并构造了方程的守恒律。
Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations
研究无散射Kadomtsev-Petviashvili方程在不同初始数据下的奇异形成，利用傅里叶系数渐进行为确定临界时间和位置及奇异类型，验证Hopf方程解破裂的准确性，并分析Burgers方程和Korteweg-de Vries方程的解正则化情况。
Numerical study of the Kadomtsev--Petviashvili equation and dispersive shock waves
本文研究了Kadomtsev-Petviashvili I方程中色散激波的长期行为，发现调制孤子解从色散激波中出现，通过Whitham调制方程描述色散激波，KPII方程在孤子线附近为双曲型，而KPI方程为椭圆型，导致聚焦效应和孤子形成。
A plethora of fully localised solitary waves for the full-dispersion Kadomtsev-Petviashvili equation
本文研究了全 dispersion KP-I方程的完全局部化孤立波，通过扰动方法和隐函数定理变体，证明其存在对称的孤立波家族。
Shock formation in the dispersionless Kadomtsev-Petviashvili equation
无色散Kadomtsev-Petviashvili（dKP）方程$(u_t+uu_x)_x=u_{yy}$是描述二维激波的最简单的非线性波方程之一。为了求解dKP方程，我们使用了一个坐标变换的启发，一维的霍普夫方程$u_t+uu_x=0$的特征方法。我们数值表明，转换后的方程的解决方案不发展冲击。这使我们能够扩展dKP解决方案作为一个多值函数的图形超过临界时间时，梯度爆炸。在$(x,y)$平面上的唇形区域中，这个翻转解是多值的，在这个区域中，dKP方程的解只在弱意义下存在，并且形成了激波阵面。局部展开揭示了激波的普遍标度结构，在适当改变坐标后，它对应于一个通用的尖点突变。我们提供了一个启发式推导的冲击波阵面的位置附近的临界点的dKP方程的解决方案，并研究了dKP方程的解决方案时，添加少量的耗散。利用多尺度分析，我们证明了在小耗散的极限和dKP解的临界点附近，耗散dKP方程的解收敛到一个Pearcey积分.我们测试和说明我们的结果与数值模拟的正则化方程，dKP方程，并给出的Pearcey积分的渐近描述的详细比较。
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