A. Boutet de Monvel, J. Lenells, D. Shepelsky
We analyze the long-time asymptotics for the Degasperis--Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated $3 \times 3$-matrix valued Riemann--Hilbert problem, we find an explicit formula for the leading order asymptotics of the solution in the similarity region in terms of the initial and boundary values.
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We analyze a class of initial-boundary value problems for the Degasperis-Procesi equation on the half-line. Assuming that the solution $u(x,t)$ exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $k$.
We consider the general Degasperis-Procesi model of shallow water out-flows. This six parametric family of conservation laws contains, in particular, KdV, Benjamin-Bona-Mahony, Camassa-Holm, and Degasperis-Procesi equations. The main result consists of criterions which guarantee the existence of solitary wave solutions: solitons and peakons ("peaked solitons").
We prove that the only global, strong, spatially periodic solution to the Degasperis-Procesi equation, vanishing at some point (t0, x0), is the identically zero solution. We also establish the analogue of such Liouville-type theorem for the Degasperis-Procesi equation with an additional dispersive term.
We consider the general Degasperis-Procesi model of shallow water out-flows. This fife parametric family of conservation laws contains, in particular, KdV, Camassa-Holm, and Degasperis-Procesi equations. The main result consists of a criterion which guarantees the existence of a smooth soliton type solution. We discuss also the scenario of soliton interaction for this model in the nonintegrable case.
In this paper, we prove an orbital stability result for the Degasperis-Procesi peakon with respect to perturbations having a momentum density that is first negative and then positive. This leads to the orbital stability of the antipeakon-peakon profile with respect to such perturbations.
In the paper, we consider the Cauchy problem for a generalized Degasperis-Procesi equation. We prove that the data-to-solution map is not uniformly continuous.
In this paper, we obtain the local-in-time existence and uniqueness of solution to the Degasperis-Procesi equation in $B^1_{\infty,1}(\R)$. Moreover, we prove that the data-to-solution of this equation is continuous but not uniformly continuous in $B^1_{\infty,1}$.
The Degasperis-Procesi equation is an approximating model of shallow-water wave propagating mainly in one direction to the Euler equations. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations with the same asymptotic accuracy, and is integrable with the bi-Hamiltonian structure. In the present study, we establish existence and spectral stability results of localized smooth solitons to the Degasperis-Procesi equation on the real line. The stability proof relies essentially on refined spectral analysis of the…
This paper is concerned with global existence and blow-up phenomena for two-component Degasperis-Procesi system and two-component b-family system. The strategy relies on our observation on new conservative quantities of these systems. Several new global existence results and a new blowup result of strong solutions to the two-component Degasperis- Procesi system and the two-component b-family system are presented by using these new conservative quantities.
In this paper, we study the Cauchy problem for the two component Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$. By presenting a new construction of initial data, we proved the norm inflation of the corresponding solutions in $B^1_{\infty,1}(\mathbb R)$ and hence ill-posedness. This is quite different from the local well-posedness result for the Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$ due to the coupled structure of density function.
This paper is concerned with the Cauchy problem for a two-component Degasperis-Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented.
In this paper, we present a new argument (see Lemma 3.4) that allows us to simplify the proof of stability of peakons established in Lin and Liu (2009) (Theorem 1.1).
We present an inverse scattering transform approach to the Cauchy problem on the line for the Degasperis--Procesi equation $u_t-u_{txx}+3ωu_x+4uu_x=3u_xu_{xx}+uu_{xxx}$ in the form of an associated Riemann-Hilbert problem. This approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used in studying its long-time behavior.
We show that the weakly dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton, and Novikov equations can be reduced to their non-dissipative versions by means of an exponentially time-dependent scaling. Hence, up to a simple change of variables, the non-dissipative and dissipative versions of these equations are equivalent. Similar results hold also for the equations in the so-called b-family of equations as well as for the two-component and μ-versions of the above equations.
We consider the non-local formulation of the Degasperis-Procesi equation $u_t+uu_x+L(\frac{3}{2}u^2)_x=0$, where $L$ is the non-local Fourier multiplier operator with symbol $m(ξ)=(1+ξ^2)^{-1}$. We show that all $L^\infty$, pointwise travelling-wave solutions are bounded above by the wave-speed and that if the maximal height is achieved they are peaked at those points, otherwise they are smooth. For sufficiently small periods we find the highest, peaked, travelling-wave solution as the limiting case at the end of the main bifurcation curve of $P$-periodic solutions. The results imply that the …
We improve the decay argument by [Bona and Li, J. Math. Pures Appl., 1997] for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady Degasperis-Procesi equation in the nonlocal formulation. In addition, we give a method which confirms the symmetry of solitary waves of the maximum height. Finally, we discover how the symmetric structure is connected to the steady structure of solutions to the Degasperis-Procesi equation, and give a more intuitive proof for symmetric solutions to be traveling waves. The improved argume…
We prove that the peakons are asymptotically H 1-stable, under the flow of the Degasperis-Procesi equation, in the class of functions with a momentum density that belongs to M + (R). The key argument is a rigidity result for uniformly in time exponentially decaying global solutions that is shared by the Holm-Staley b-family of equations for b $\ge$ 1. This extends previous results obtained for the Camassa-Holm equation (b = 2).
Stability of peakons for the Degasperis-Procesi equation
Degasperis-Procesi方程可以作为Euler方程的一个单参数渐近浅水近似族的一员导出，其渐近精度与Camassa-Holm方程相同。本文研究了线上Degasperis-Procesi方程峰值孤子的轨道稳定性问题。通过构造一个李雅普诺夫函数，我们证明了这些峰孤子的形状在小扰动下是稳定的。
We consider the dispersive Degasperis-Procesi equation $u_t-u_{x x t}-\mathtt{c} u_{xxx}+4 \mathtt{c} u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x=0$ with $\mathtt{c}\neq 0$. In \cite{Deg} the authors proved that this equation possesses infinitely many conserved quantities. We prove that, in a neighborhood of the origin, there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of some $H^s$ Sobolev space, both on $\mathbb{R}$ and $\mathbb{T}$. By the analysis of these conserved quantities we deduce a result of global wel…
In this paper, a two-component variant of the Degasperis-Procesi equation on the real line is discussed. Applying Kato's theory, we first prove the local well-posedness for the equation under consideration in $H^s\times H^{s-1}$, for $s\geq 2$. Second we establish the precise blow-up scenario. For compactly supported initial data, we show that the associated solution does not have compact support for any positive time; the localized initial disturbance propagates with an infinite speed. Although the solution is no longer compactly supported we prove that it decays at an exponentially fast rate…
We use geometric methods to study two natural two-component generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle $\Diff(S^1)$ with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm e…
The Degasperis-Procesi equation possesses well-known peaked solitary waves that are called peakons. Their stability has been established by Lin and Liu in [5]. In this paper, we localize the proof (in some suitable sense detailed in Section 3) of the stability of a single peakon. Thanks to this, we extend the result of stability to the sum of N peakons traveling to the right with respective speeds c1, . . . , cN , such that the difference between consecutive locations of peakons is large enough.
Highest waves for fractional Korteweg--De Vries and Degasperis--Procesi equations
研究分数阶KdV和DP方程的行波，证明存在从常数解分叉的局部解析分支并延伸至全局，极限得到最高尖顶波，且具有最优s-Hölder正则性。
We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis-Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash-Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special w…
The Degasperis-Procesi equation is the integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the Desgasperis-Procesi (DP) equation on the real line. %extending our previous work on their spectral stability \cite{LLW}. The main difficulty stems from the fact that the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in th…
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
近年来，两类非线性色散偏微分方程由于其可积结构而引起了广泛的关注。我们证明，这两个方程出现在浅水波的传播模型在一个平坦的床。该方程比经典的非线性色散Benjamin-Bona-Mahoney方程和Korteweg-de Vries方程具有更强的非线性效应。特别是，它们适应波浪破碎现象。
Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation
本文研究了具有Dirichlet边界条件的有界区域中的带源项的守恒律。我们首先证明了边界处强迹的存在，以提供一个简单的熵边界条件。配备了这个公式，我们继续建立熵的初边值问题的解的适定性。证明利用动力学公式和补偿紧性方法。最后，我们利用这些结果证明了Degasperis-Procesi浅水方程初边值问题的一类间断解的适定性，该方程是一个三阶非线性色散方程，可以改写为具有非局部源项的非线性守恒律的形式.
Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are…
The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the $L^2\cap L^\infty$ orbital stability of a wave train containing $N$ smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the $L^2$-norm, which by itself can not bound the higher-ord…
Well-posedness of the Stochastic Degasperis-Procesi Equation
本文利用动力学理论，在 $L^2(\mathbb{R})\cap L^{2+δ}(\mathbb{R})$ 初值条件下证明了带加性噪声的随机 Degasperis-Procesi 方程具有全局路径解，并在零噪声情形下改进了已有确定性结果。
We derive a precise energy stability criterion for smooth periodic waves in the Degasperis--Procesi (DP) equation. Compared to the Camassa-Holm (CH) equation, the number of negative eigenvalues of an associated Hessian operator changes in the existence region of smooth perodic waves. We utilize properties of the period function with respect to two parameters in order to obtain a smooth existence curve for the family of smooth periodic waves of a fixed period. The energy stability condition is derived on parts of this existence curve which correspond to either one or two negative eigenvalues of…
We consider the periodic $\muDP$ equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection $\nabla$ on the Fréchet Lie group $\Diff^{\infty}(§^1)$ of all smooth and orientation-preserving diffeomorphisms of the circle $§^1=\R/\Z$. On the Lie algebra $\C^{\infty}(§^1)$ of $\Diff^{\infty}(§^1)$, this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of $\muDP$ which depends smoothly on time and on the initial data. …
Finite-time blow-up for the Geng--Xue system with an application to the three-component Degasperis--Procesi system
本文研究了三组分Degasperis-Procesi方程的初值问题，探讨了低Besov空间下的局部解存在性及爆发准则，并通过无需守恒律的方法研究了爆发现象及解的持久性。
Local isometric immersions and breakdown of manifolds determined by Cauchy problems of the Degasperis-Procesi equation
本研究证明Degasperis-Procesi方程的局部与非局部形式均具伪球面性质，明确了其柯西问题解定义的伪球面度量正交余标架、第二基本形式的局部存在性，给出波破碎时曲面坍缩的结论与余标架在上半平面有定义的条件，并探究相关可积结构。
We prove reducibility of a class of quasi-periodically forced linear equations of the form \[ \partial_tu-\partial_x\circ (1+a(ωt, x))u+\mathcal{Q}(ωt)u=0,\quad x\in\mathbb{T}:=\mathbb{R}/2π\mathbb{Z}, \] where $u=u(t,x)$, $a$ is a small, $C^{\infty}$ function, $\mathcal{Q}$ is a pseudo differential operator of order $-1$, provided that $ω\in\mathbb{R}^ν$ satisfies appropriate non-resonance conditions. Such PDEs arise by linearizing the Degasperis-Procesi (DP) equation at a small amplitude quasi-periodic function. Our work provides a first fundamental step in developing a KAM theory for pertur…
Weak solution for Stochastic Degasperis-Procesi Equation
针对带无穷维乘性噪声的随机 Degasperis-Procesi 方程，论文结合动力学理论与随机紧性证明弱动力学鞅解存在，并由轨道唯一性得到概率意义下的强解。
Linear Asymptotic Stability of the Smooth 1-Solitons for the Degasperis-Procesi Equation
本文研究了Degasperis-Procesi方程中光滑1孤子的渐近稳定性，利用方程的完全可积性证明了光滑孤立波的强谱稳定性，并通过谱间隙结果得到指数衰减估计，最终在指数加权空间中建立了线性渐近稳定性结果。
In this article, we study the self-similar solutions of the 2-component Degasperis-Procesi water system:% [c]{c}% ρ_{t}+k_{2}uρ_{x}+(k_{1}+k_{2})ρu_{x}=0 u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}+k_{3}ρρ_{x}=0. By the separation method, we can obtain a class of self-similar solutions,% [c]{c}% ρ(t,x)=\max(\frac{f(η)}{a(4t)^{(k_{1}+k_{2})/4}},\text{}0),\text{}u(t,x)=\frac{\overset{\cdot}{a}(4t)}{a(4t)}x \overset{\cdot\cdot}{a}(s)-\fracξ{4a(s)^κ}=0,\text{}a(0)=a_{0}% \neq0,\text{}\overset{\cdot}{a}(0)=a_{1} f(η)=\frac{k_{3}}ξ\sqrt{-\fracξ{k_{3}}η^{2}+(\fracξ{k_{3}}α) ^{2}}% where $η=\frac{x}{a…
The Cauchy problem for the Degasperis-Procesi Equation: Painlevé Asymptotics in Transition Zones
本文研究 Degasperis-Procesi 方程柯西问题在孤立子区域与无孤立子区域之间两个过渡区中的长时间渐近，利用 Painlevé II 方程的解给出首阶近似，方法结合 Deift-Zhou 非线性最速下降法的 $\bar \partial$-推广与双尺度极限技巧。
On the rotation-two-component $b$-family shallow-water system: derivation and wave-breaking analysis
本文从Green--Naghdi型方程推导旋转双分量$b$-族浅水波系统，引入辅助变量建立局部$L^\infty$估计，克服无守恒量困难，给出两个爆破准则，并推广至周期情形。
In this article we study a new kind of unbounded solutions to the Novikov equation, found via a Lie symmetry analysis. These solutions exhibit peakon creation, i.e., these solutions are smooth up until a certain finite time, at which a peak is created. We show that the functions are still weak solutions for those times where the peak lives. We also find similar unbounded solutions with peakon creation in the related Camassa-Holm equation, by making an ansatz inspired by the Novikov solutions. Finally, we see that the same ansatz for the Degasperis-Procesi equation yields unbounded solutions wh…
A Generalized Framework for Singular Fractional Burgers Equations with Stochastic Forcing
本文研究了由 $|D|^{1/2}ξ$ 驱动的广义分数奇 Burgers 方程的适定性，建立了 GFSB 方程框架并证明了其局部适定性，以及在特定条件下解的广义性质。
In this paper, we study a weakly dissipative variant of the periodic Degasperis-Procesi equation. We show the local well-posedness of the associated Cauchy problem in $H^s(§)$, $s>3/2$, and discuss the precise blow-up scenario for $s=3$. We also present explicit examples for globally existing solutions and blow-up.
Popowicz system, as the interacting system of Camassa-Holm and Degasperis-Procesi equations, has attracted some attention in recent years. In this paper, we first study the local well-posedness for the cauchy problem of Popowicz system in nonhomogeneous Besov spaces $B^s_{p,r}\times B^s_{p,r}$ with $s> \max\{2, \frac{1}{p}+\frac{3}{2}\}$ or $(s=2, 2\leq p \leq \infty, 1\leq r\leq 2)$. Moreover, a new blow-up criterion and global existence with different initial values are obtained.
We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $p\in [1,\infty], r\in (1,\infty]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works (\cite{Danchin2,Byers,HHK}).
This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool o…
A finite difference scheme for smooth solutions of the general mKdV equation
针对一般mKdV方程，提出一种经济的有限差分格式，在特殊条件下保持全局经典解性质，并展示孤子传播与相互作用的数值结果。
Asymptotic stability of solitary waves for the b-family of equations
本文研究了$b < -1$范围内$b$族方程中左孤立解的渐近稳定性，通过非线性Liouville性质和谱分析，首次建立了该类非可积方程中孤立波的稳定性理论。
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