On the support of solutions to the g-KdV equation

Power series solution of the modified KdV equation

We prove local-wellposedness of the mKdV equation in $\mathcal{F}L^{s,p}$ spaces using the new method of M. Christ.

A decay property of solutions to the k-generalized KdV equation

We use a Leibnitz rule type inequality for fractional derivatives to prove conditions under which a solution $u(x,t)$ of the k-generalized KdV equation is in the space $L^2(|x|^{2s}\,dx)$ for $s \in \mathbb R_{+}$.

A-priori bounds for KdV equation below $H^{-3/4}$

We consider the Korteweg-de Vries Equation (KdV) on the real line, and prove that the smooth solutions satisfy a-priori local in time $H^s$ bound in terms of the $H^s$ size of the initial data for $s\geq -4/5$.

KdV方程2-孤子轮廓的变分刻画

A variational characterization of 2-soliton profiles for the KdV equation

该研究利用轮廓分解，将KdV方程的2-孤子解刻画为涉及三个多项式守恒律的约束变分问题的全局极小值，实现了2-孤子轮廓的变分表征。

Global dynamics of the generalized fifth-order KdV equation with critical nonlinearity

We prove global existence and modified scattering for the solutions of the generalized fifth-order KdV equation with critical nonlinearity for small and localized initial data. The proof is undergoing by using the space-time resonance method and the stationary phase argument.

On the Cauchy problem for quasi-linear Hamiltonian KdV-type equations

We prove local in time well-posedness for a class of quasilinear Hamiltonian KdV-type equations with periodic boundary conditions, more precisely we show existence, uniqueness and continuity of the solution map. We improve the previous result in \cite{Mietka}, generalising the considered class of equations and improving the regularity assumption on the initial data.

On the radius of spatial analyticity for the quartic generalized KdV equation

Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Grujić and Kalisch.

Local and global well-posedness of the stochastic KdV-Burgers equation

The stochastic PDE known as the Kardar-Parisi-Zhang equation (KPZ) has been proposed as a model for a randomly growing interface. This equation can be reformulated as a stochastic Burgers equation. We study a stochastic KdV-Burgers equation as a toy model for this stochastic Burgers equation. Both of these equations formally preserve spatial white noise. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This paper establishes a result in this direction. After smoothing the additive noise (by less than one spatial derivative), we esta…

Dispersive decay of small data solutions for the KdV equation

We consider the Korteweg-de Vries (KdV) equation, and prove that small localized data yields solutions which have dispersive decay on a quartic time-scale. This result is optimal, in view of the emergence of solitons at quartic time, as predicted by inverse scattering theory.

Lower bounds on the radius of spatial analyticity for the KdV equation

We present estimates for the radius of analyticity of solutions to the Korteweg-de Vries equation, which improve earlier results due to Bona, Grujić and Kalisch.

Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation

It is shown that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the KdV equation cannot decay faster than $|t|^{-4/3}$ as $|t| \to \infty$ given initial data that is analytic with fixed radius $σ_0$. This improves a recent result of Selberg and Da Silva, where they proved a decay rate of $|t|^{-(4/3 + \varepsilon) }$ for arbitrarily small positive $\varepsilon$. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear $L^2$ estimates associated with the KdV equa…

Local well-posedness for the higher-order generalized KdV type equation with low-degree of nonlinearity

This paper is concerned with the local well-posedness for the higher-order generalized KdV type equation with low-degree of nonlinearity. The equation arises as a non-integrable and lower nonlinearity version of the higher-order KdV equation. As for the lower nonlinearity model of the KdV equation, Linares, the author and Ponce [11] prove the local well-posedness under a non-degenerate condition introduced by Cazenave and Naumkin [1]. In this paper, it turns out that the well-posedness result can be extended into the higher-order equation. We also give a lower bound for the lifespan of the sol…

New lower bounds on the radius of spatial analyticity for the KdV equation

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{-1/4}$ as time $t$ goes to infinity. This improves the works [Selberg, da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Annales Henri Poincaré, 2017, 18(3): 1009-1023] and [Tesfahun, Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation, arXiv preprint arXiv:1707.07810, 2017]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspir…

A note on the stability for Kawahara-KdV type equations

In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara-KdV equation $$u_t+uu_x+u_{xxx}-γ_1 u_{xxxxx}=0,$$ and the modified Kawahara-KdV equation $$u_t+3u^2u_x+u_{xxx}-γ_2 u_{xxxxx}=0,$$ where $γ_i\in\mathbb{R}$ is a positive number when $i=1,2$. The main approach used to determine the stability of solitary traveling-waves will be the theory developed by Albert

Variable depth KDV equations and generalizations to more nonlinear regimes

We study here the water-waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known, that for such regimes, a generalization of the KdV equation (somehow linked to the Camassa-Holm equation) can be derived and justified by A. Constantin, D. Lannes "The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations" when the bottom is flat. We generalize here this result with a new class of equations taking into account variable bottom topographies. Of course, the many variabl…

A sharp condition for the well-posedness of the linear KdV-type equation

To make the illposedness argument more transparent the argument is rewritten to reduce the equation to the constant dispersion case. Minor errors are corrected. Accepted to the Proceedings of the AMS.

A uniqueness result for 2-soliton solutions of the KdV equation

Multisoliton solutions of the KdV equation satisfy nonlinear ordinary differential equations which are known as stationary equations for the KdV hierarchy, or sometimes as Lax-Novikov equations. An interesting feature of these equations, known since the 1970's, is that they can be explicitly integrated, by virtue of being finite-dimensional completely integrable Hamiltonian systems. Here we use the integration theory to investigate the question of whether the multisoliton solutions are the only nonsingular solutions of these ordinary differential equations which vanish at infinity. In particul…

关于无限项链图上Boussinesq方程的KdV近似

On the KdV approximation for a Boussinesq equation posed on the infinite necklace graph

该研究针对无限周期项链图上的Boussinesq方程，推导用于描述长波行波的KdV方程，通过能量估计给出误差界，验证了该形式近似的有效性。

A note on ill-posedness for the KdV equation

We prove that the solution-map $ u_0 \mapsto u $ associated with the KdV equation cannot be continuously extended in $ H^s(\R) $ for $ s<-1 $. The main ingredients are the well-known Kato smoothing effect for the mKdV equation as well as the Miura transform.

Weak Continuity of Dynamical Systems for the KdV and mKdV Equations

In this paper we study weak continuity of the dynamical systems for the KdV equation in H^{-3/4}(R) and the modified KdV equation in H^{1/4}(R). This topic should have significant applications in the study of other properties of these equations such as finite time blow-up and asymptotic stability and instability of solitary waves. The spaces considered here are borderline Sobolev spaces for the corresponding equations from the viewpoint of the local well-posedness theory. We first use a variant of the method of [5] to prove weak continuity for the mKdV, and next use a similar result for a mKdV…

A note on local well-posedness of generalized KdV type equations with dissipative perturbations

In this note we report local well-posedness results for the Cauchy problems associated to generalized KdV type equations with dissipative perturbation for given data in the low regularity $L^2$-based Sobolev spaces. The method of proof is based on the {\em contraction mapping principle} employed in some appropriate time weighted spaces.

Sharp local well-posedness of KdV type equations with dissipative perturbations

In this work, we study the initial value problems associated to some linear perturbations of KdV equations. Our focus is in the well-posedness issues for initial data given in the $L^2$-based Sobolev spaces. We derive bilinear estimate in a space with weight in the time variable and obtain sharp local well-posedness results.

On symmetries of KdV-like evolution equations

The $x$-dependence of the symmetries of (1+1)-dimensional scalar translationally invariant evolution equations is described. The sufficient condition of (quasi)polynomiality in time $t$ of the symmetries of evolution equations with constant separant is found. The general form of time dependence of the symmetries of KdV-like non-linearizable evolution equations is presented.

Global well-posedness and scattering for the defocusing, mass - critical generalized KdV equation

In this paper we prove that the defocusing, mass - critical generalized KdV initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R})$. We prove this via a concentration compactness argument.

Long-time behavior of solutions to the fifth-order modified KdV-type equation

We consider the long-time behavior of solutions to the fifth-order modified KdV-type equation. Using the method of testing by wave packets, we prove the small-data global existence and modified scattering. We derive the leading asymptotic in both the self-similar and oscillatory regions.

Lower Bounds for Non-Trivial Traveling Wave Solutions of Equations of KdV Type

We prove that if a solution of an equation of KdV type is bounded above by a traveling wave with an amplitude that decays faster than a given linear exponential then it must be zero. We assume no restrictions neither on the size nor in the direction of the speed of the traveling wave.

Complex-valued Burgers and KdV-Burgers equations

Spatially periodic complex-valued solutions of the Burgers and KdV-Burgers equations are studied in this paper. It is shown that for any sufficiently large time T, there exists an explicit initial data such that its corresponding solution of the Burgers equation blows up at T. In addition, the global convergence and regularity of series solutions is established for initial data satisfying mild conditions.

Global rough solutions to the critical generalized KdV equation

We prove that the initial value problem (IVP) for the critical generalized KdV equation $u_{t}+u_{xxx}+(u^5)_{x}=0$ on the real line is globally well-posed in $H^{s}(\R)$ provided $s>3/5$.

Non dispersive solutions of the generalized KdV equations are typically multi-solitons

We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense (in the spirit of [18]) and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non-dispersion.

Modified energies for the periodic generalized KdV equation and applications

We construct modified energies for the generalized KdV equation. As a consequence, we obtain quasi-invariance of the high order Gaussian measures along with $L^p$ regularity on the corresponding Radon-Nykodim density, as well as new bounds on the growth of the Sobolev norms of the solutions.

Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case

We prove that the KdV-Burgers is globally well-posed in $ H^{-1}(\T) $ with a solution-map that is analytic from $H^{-1}(\T) $ to $C([0,T];H^{-1}(\T))$ whereas it is ill-posed in $ H^s(\T) $, as soon as $ s<-1 $, in the sense that the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\T) $ to even ${\cal D}'(\T) $ at any fixed $ t>0 $ small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dissipation part of the KdV-Burgers equation allows to lower the $ C^\infty $ critical index with respect to the KdV equation, it does not permit to imp…

Almost sharp wave kinetic theory of multidimensional KdV type equations with $d\ge 3$

In this work, we study the random series expansion of a multidimensional KdV type equation with a diffusion term, the so-called Zakharov-Kuznetsov (ZK) equation. We impose random initial data and periodic boundary condition with period $L$ on this equation. Using the random series expansion, we derive the $3$-wave kinetic equation on the inertial range for $t\lesssim L^{-\varepsilon}T_{\text{kin}}$. Our result reaches kinetic time scale up to $\varepsilon$ loss.

Operator splitting for the KdV equation

We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+B(u)$ where $A$ is a linear operator and $B$ is quadratic. A particular example is the Korteweg-de Vries (KdV) equation $u_t-u u_x+u_{xxx}=0$. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.

Nonexistence of soliton-like solutions for defocusing generalized KdV equations

In this short note, we consider the global dynamics of the defocusing generalized KdV equations: u_t + u_{xxx} = (|u|^{p-1}u)_x. We use Tao's theorem that the energy moves faster than mass to prove a moment type dispersion estimate. As an application of the dispersion estimate, we show that there is no soliton-like solutions with decaying assumption.

Spectral stability of nonlinear waves in KdV-type evolution equations

This paper concerns spectral stability of nonlinear waves in KdV-type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a finite number of negative eigenvalues and an unbounded non-invertible symplectic operator $\partial_x$. The instability index theorem is proven under a generic assumption on the self-adjoint operator both in the case of solitary waves and periodic waves. This result is reviewed in the context of other recent results on spectral stability of nonlinear waves in KdV-type evolution equations.

On the fifth order KdV equation: local well-posedness and lack of uniform continuity of the solution map

In this paper we prove that the fifth order equation arising from the KdV hierarchy $ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $ is locally well-posed in $ H^s(\mathbb{R}) $ for $ s> 5/2. Also, we prove the solution map of the equation is not uniformly continuous for $s>0$.

Well-posedness of generalized KdV and one-dimensional fourth-order derivative nonlinear Schrödinger equations for data with an infinite $L^2$ norm

We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schrödinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of modulation spaces is shown, which contain some data with infinite $L^{2}$ norm.

Corrections to the KdV approximation for water waves

In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone.

Self-Similar Solutions of the Non-Strictly Hyperbolic Whitham Equations for the KdV Hierarchy

We study the Whitham equations for all the higher order KdV equations. The Whitham equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solution of the Whitham equations when the initial values are given by a step function.

Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation

The Virasoro-Bott group endowed with the right-invariant $L^2$-metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

Discrete Fourier restriction associated with KdV equations

In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term $ F(u)\p_x u$ provided $F\in C^5$ and the initial data $ϕ\in H^s$ with $s>1/2$.

Large Time Behavior of the Zero Dispersion Limit of the Fifth Order KdV Equation

We study the zero dispersion limit of the fifth order KdV equations when time is sufficiently large. In general, the weak limit may be described by an arbitrary odd number of hyperbolic equations. Unlike the KdV case, these are non-strictly hyperbolic equations. However, we show that the weak limit is governed by three hyperbolic equations in a domain in the space-time for all times bigger than a large time. Outside this domain, the weak limit satisfies a single hyperbolic equation.

Existence of Bound and Ground States for a System of Coupled Nonlinear Schrödinger-KdV Equations

We prove the existence of bound and ground states for a system of coupled nonlinear Schrödinger-Korteweg-de Vries equations, depending on the size of the coupling coefficient.

Characterization of Rational Solutions of a KdV-Like Equation

We characterize the rational solutions to a KdV-like equation which are generated from polynomial solutions to the corresponding generalized bilinear equation. We use a particular class of polynomials satisfying a quadratic difference equation to obtain that there are no solutions of the bilinear equation with degree (in the spatial variable) greater than 5. As a byproduct, we answer positively a conjecture of Yi Zhang and Wen-Xiu Ma about these solutions.

Cost for a controlled linear KdV equation

The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.

Inverted solutions of KdV-type and Gardner equations

In most of the studies concerning nonlinear wave equations of Korteweg-de Vries type, the authors focus on waves of elevation. Such waves have general form ~$u_{\text{u}}(x,t)=A f(x-vt)$, where ~$A>0$. In this communication we show that if ~$u_{\text{up}}(x,t)=A f(x-vt)$ is the solution of a given nonlinear equation, then $u_{\text{down}}(x,t)=-A f(x-vt)$, that is, an inverted wave is the solution of the same equation, but with changed sign of the parameter ~$α$. This property is common for KdV, extended KdV, fifth-order KdV, Gardner equations, and generalizations for cases with an uneven bott…

Normal form coordinates for the KdV equation having expansions in terms of pseudodifferential operators

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudodifferential operator of order 0 with principal part given by the Fourier transform and (2) the pullback of the KdV Hamiltonian is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the …

具有$L^2$数据的KdV和修正KdV方程的有限差分方法的收敛性

Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data

针对KdV和修正KdV方程，提出一种带四阶稳定项的半离散有限差分方法，证明在$L^2$非光滑数据下强收敛，并通过数值实验验证。

Singular Miura type initial profiles for the KdV equation

We show that the KdV flow evolves any real singular initial profile q of the form q=r'+r^2, where r\inL_{loc}^2, r|_{R_+}=0 into a meromorphic function with no real poles.

A Mathematical Justification of the Momentum Density Function Associated to the KdV Equation

Consideration is given to the KdV equation as an approximate model for long waves of small amplitude at the free surface of an inviscid fluid. It is shown that there is an approximate momentum density associated to the KdV equation, and the difference between this density and the physical momentum density derived in the context of the full Euler equations can be estimated in terms of the long-wave parameter.

On classical solutions of the KdV equation

\begin{abstract} We show that if the initial profile $q\left( x\right) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $\int^{\infty }x^{5/2}\left\vert q\left( x\right) \right\vert dx<\infty,$ (no decay at $-\infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. \end{abstract}

Semiclassical limit for generalized KdV equations before the gradient catastrophe

We study the semiclassical limit of the (generalised) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in $H^s$ to the solution of the Hopf equation, provided the initial data belongs to $H^s$, ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities.

KdV-type equations linked via Baecklund transformations: remarks and perspectives

Third order nonlinear evolution equations, that is the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other ones are considered: they all are connected via Baecklund transformations. These links can be depicted in a wide Baecklund Chart} which further extends the previous one constructed in [22]. In particular, the Baecklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Baecklund Chart. That …

On integrability of a (2+1)-dimensional perturbed Kdv equation

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé test for integrability well, and its 4$\times $4 Lax pair with two spectral parameters is found. The results show that the Painlevé classification of coupled KdV equations by A. Karasu should be revised.