J. Colliander, C. Kenig, G. Staffilani
Oscillatory integral techniques are used to study the well-posedness of the KP-I equation for initial data that are small with respect to the norm of a weighted Sobolev space involving derivatives of total order no larger than 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 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 $N \geq 2$.
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 aim of this paper is to prove various ill-posedness and well-posedness results on the Cauchy problem associated to a class of fractional Kadomtsev-Petviashvili (KP) equations including the KP version of the Benjamin-Ono and Intermediate Long Wave equations.
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.
The Cauchy-problem for the generalized Kadomtsev-Petviashvili-II equation $$u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3,$$ is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines local smoothing and maximal function estimates as well as bilinear refinements of Strichartz type inequalities via multilinear interpolation in $X_{s,b}$-spaces.
We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I equation which vanish for $x\to-\infty$ and study their large time asymptotic behavior. We prove that such solutions eject (for $t\to\infty$) a train of curved asymptotic solitons which move behind the basic wave packet.
In this paper we establish some results concerning the existence, regularity and concentration phenomenon of nontrivial solitary waves for a Generalized Kadomtsev-Petviashvili (GKP) equation in $\mathbb{R}^2.$ Variational methods are used to get an existence result and to study the concentration phenomenon, while the regularity is more delicate because we are leading with functions in an anisotropic Sobolev space.
In this paper we obtain local in time existence and (suitable) uniqueness and continuous dependence for the KP-I equation for small data in the intersection of the energy space and a natural weighted $L^{2}$ space.
In this paper we establish the local and global well-posedness of the real valued fifth order Kadomstev-Petviashvili I equation in the anisotropic Sobolev spaces with nonnegative indices. In particular, our local well-posedness improves Saut-Tzvetkov's one and our global well-posedness gives an affirmative answer to Saut-Tzvetkov's $L^2$-data conjecture.
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…
We prove local and global well-posedness in $H^{s,0}(\mathbb{R}^{2})$, $s > -1/2$, for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers-I equation (KPBI) by working in Bourgain's type spaces. This result is almost sharp if one requires the flow-map to be smooth.
We consider the orbital stability of solitons of the Kadomtsev--Petviashvili-I equation in $\mathbb{R} \times (\mathbb{R}/2π\mathbb{Z})$ which is one of a high dimensional generalization of the Korteweg--de Vries equation. Benjamin showed that the Korteweg--de Vries equation possesses the stable one soliton. We regard the one soliton of the Korteweg--de Vries equation as a line soliton of the Kadomtsev--Petviashvili-I equation. Zakharov and Rousset--Tzvetkov proved the orbital instability of the line solitons of the Kadomtsev--Petviashvili-I equation on $\mathbb{R}^2$. In the case of the Kadom…
We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space H^{s_1,s_2}(R^2) with s_1 > -1/2 and s_2 \geq 0. On the H^{s_1,0}(R^2) scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for a dispersion generalised KP II type equation. We also deduce a global well-posedness result for the generalised equation.
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。
We first review the known mathematical results concerning the KP type equations. Then we perform numerical simulations to analyze various qualitative properties of the equations : blow-up versus long time behavior, stability and instability of solitary waves.
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。
We numerically investigate transverse stability and instability of so-called cnoidal waves, i.e., periodic traveling wave solutions of the Korteweg-de Vries equation, under the time-evolution of the Kadomtsev-Petviashvili equation. In particular, we find that in KP-I small amplitude cnoidal waves are stable (at least for spatially localized perturbations) and only become unstable above a certain threshold. In contrast to that, KP-II is found to be stable for all amplitudes, or, equivalently, wave speeds. This is in accordance with recent analytical results for solitary waves given in \cite{RT1…
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。
An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev--Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painleve' I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture.
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。
We prove global well-posedness for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers equation (KPBII) in $\mathbb R^2$ when the initial value belongs to the anisotropic Sobolev space $H^{s_1,s_2}(\mathbb R^2)$ for all $s_1>-\frac12$ and $s_2\geq0$. On the other hand, we prove in some sense that our result is sharp.
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 goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \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*} is locally well-posed in the anisotropic Sobolev spaces$ H^{s_{1},\>s_{2}}(\R^{2})$ with $s_{1}>-\frac{α-1}{4}$ and $s_{2}\geq 0$. Secondly, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{(α-1)(3α-4)}{4(5α+3)}$ if $4\leq α\leq5$. Finally, we prove that the problem is globally well-posed in $H^{s_{1},\>0…
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.
The (2+1)-dimensional spherical Kadomtsev-Petviashvili (SKP) equation of J.-K. Xue [Phys. Lett. A 314:479-483 (2003)] fails the Painleve test for integrability at the highest resonance, where a nontrivial compatibility condition for recursion relations appears. This compatibility condition, however, is sufficiently weak and thus allows the SKP equation to possess an integrable (1+1)-dimensional reduction, which is detected by the Weiss method of truncated singular expansions.
In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable struc…
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 …
Equivalence is established between a special class of Painleve VI equations parametrized by a conformal dimension $μ$, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give rational solutions of the Painleve VI equation for $μ=1,2,{...} $.
A new method for the solution of initial-boundary value problems for \textit{linear} and \textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 \cite{F1997}. This approach was subsequently extended to initial-boundary value problems for evolution PDEs in two spatial dimensions, first in the case of linear PDEs \cite{F2002b} and, more recently, in the case of integrable nonlinear PDEs, for the Davey-Stewartson and the Kadomtsev-Petviashvili II equations on the half-plane (see \cite{FDS2009} and \cite{MF2011} respectively). In this work…
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…
The KP-I equation \[ (u_t-2uu_x+\tfrac{1}{2}(β-\tfrac{1}{3})u_{xxx})_x -u_{yy}=0 \] arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number $β>1/3$). This equation admits --- as an explicit solution --- a `fully localised' or `lump' solitary wave which decays to zero in all spatial directions. Recently there has been interest in the \emph{full-dispersion KP-I equation} \[u_t + m({\mathrm D}) u_x + 2 u u_x = 0,\] where $m({\mathrm D})$ is the Fourier multiplier with symbol \[ m(k) = \left( 1 + β|k|^2|\right)^{\frac{1}{2}} \left( \frac{\ta…
We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as $\partial_tu+α\partial_x^3u+\partial^5_xu+\partial_x^{-1}\partial_y^2u+uu_x=0,$ while $α\in \mathbb{R}$. We introduce an interpolated energy space $E_s$ to consider the well-posedeness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in $E_s$ for $0<s\leq1$. To obtain the local well-posedness, we present a bilinear estimate in the Bourgain space in the framework of the interpolated energy space. It crucially depends on the dyadic…
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。
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 KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics where it describes waves in shallow water. It provides a multidimensional generalisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey-Stewartson equation \cite{FDS2009}, in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called d-bar…
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。
The dispersionless Kadomtsev-Petviashvili (dKP) equation $(u_t+uu_x)_x=u_{yy}$ is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation $u_t+uu_x=0$. We show numerically that the solutions to the transformed equation do not develop shocks. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape reg…
Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations
本文证明非交换KP方程与“提升”mKP方程可直接线性化且可积，推导了“提升”mKP方程及对数势形式，构造预Poppe代数并通过多项式展开验证可积性，还对类孤子相互作用做了数值模拟。
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