The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{1/4} (\R)$ endpoint for mKdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV 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.
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.
In this paper we study the generalized Korteweg de Vries (KdV) equation with the nonlinear term of order three: $(u^{3+1})_x$. We prove sharp local well--posedness for the initial and boundary value problem posed on the right half line. We thus close the gap in the well--posedness theory of the generalized KdV which remained open after the seminal work of Colliander and Kenig in \cite{CK}.
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.
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.
We prove the sharp global well-posedness results for the initial value problems (IVPs) associated to the modified Korteweg-de Vries (mKdV) equation and a system modeled by the coupled modified Korteweg-de Vries equations (mKdV-system). To obtain our results we use the second generation of the modified energy and almost conserved quantities, more precisely, the refined I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.
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.
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…
This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right. \end{equation*} where $L_p$ is a dissipative multiplicator operator. Using Besov-Bourgain Spaces, we establish a bilinear estimate and following the framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global well-posedness in the Sobolev spaces $H^{-p/2}(I\!\!R)$ and sharp ill-posedness in $H^s(I\!\!R)$ when $s<-p/2$ with $p \geq 2$.
The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ are determined. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $-\infty < x < \infty$, exponentially decreasing to zero as $|x| \to \infty$, that blow up in finite time.
The Cauchy problem for the modified KdV equation is shown to be locally well posed for data u_0 in the space \hat(H^r_s) defined by the norm ||u_0||:=||<ξ>^s \hat(u_0)||_L^r', provided 4/3 < r \le 2, s \ge 1/2 - 1/(2r). For r=2 this coincides with the best possible result on the H^s - scale due to Kenig, Ponce and Vega. The proof uses an appropriate variant of the Fourier restriction norm method and linear as well as bilinear estimates for the solutions of the Airy equation.
In this paper, we consider the fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in $H^s(\mathbb T)$, $s > 2$, via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173 (2) (2008) 265--304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estim…
We establish a new a priori bound for $ L^2 $-bounded sequences of solutions to the mKdV equations on the torus. This first enable us to construct weak solutions in $ L^2$ for this equation and to check that the "solutions" constructed by Kappeler and Topalov in the defocusing case satisfy the equation in some weak sense. In a second time, we prove that the solution-map associated with the mKdV and the KdV equation are discontinuous for the $ H^s(\T) $ topology for respectively $ s<0$ and $ s<-1$. These last results are sharp.
The long time behavior of solutions to the defocusing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through $\overline{\partial}$-derivatives. To extend the asymptotics to solutions with initial data in lower regularity spaces, we apply a global approximation via PDE techniques.
We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness result by showing that the flow-map is not continuous if the given data has Sobolev regularity $s< 1$. The results obtained in this work are sharp.
We study the Cauchy problem for the modified KdV equation for data u_0 in the space ^H^r_s defined by the norm ||u_0||_{^H^r_s}:=||<ξ>^s u^_0||_{L^r'_ξ}. Local well-posedness of this problem is established in the parameter range 2>=r>1, s>=1/2-1/2r, so the case (s,r)=(0,1), which is critical in view of scaling considerations is almost reached. To show this result, we use an appropriate variant of the Fourier restriction norm method as well as bi- and trilinear estimates for solutions of the Airy equation.
This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified …
We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $2\leq p < \infty$. For $s < \frac 14$, we show that the solution map for mKdV is not locally uniformly continuous in $M^{2,p}_{s}(\mathbb{R})$. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $…
Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann--Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified Korteweg-de Vries equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of $I$-method, we give its global well-posedness in lower regularity Sobolev spaces, and then obtain the asymptotic behavior in these spaces with weights.
In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begin{eqnarray*} u_{t}+(-1)^{j+1}\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2}) = 0,j\in N^{+},x\in\mathbf{T}= [0,2πλ) \end{eqnarray*} with low regularity data and $λ\geq 1$. Firstly, we show that the Cauchy problem for the periodic higher-order KdV equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -j+\frac{1}{2},j\geq2.$ By using some new Strichartz estimate and some new function spaces, we also show that the Cauchy problem for the periodic higher-order KdV equation is ill-posed in …
Bourgain(1993) proved that the periodic modified KdV equation (mKdV) is locally well-posed in Sobolev spave H^s(T), s >= 1/2, by introducing new weighted Sobolev spaces X^s,b, where the uniqueness holds conditionally, namely in the intersection of C([0, T]; H^s) and X^s,b. In this paper, we establish unconditional well-posedness of mKdV in H^s(T), s >= 1/2, i.e. we in addition establish unconditional uniqueness in C([0, T]; H^s), s >= 1/2, of solutions to mKdV. We prove this result via differentiation by parts. For the endpoint case s = 1/2, we perform careful quinti- and septi-linear estimate…
The soliton resolution for the focusing modified Korteweg-de vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation through $\bar{\partial}$-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory $x=\textrm{v}t$ for any fixed $ \textrm{v} $. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As byproducts of our long-time asymptotics, we also o…
We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations.
We complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $ H^{-1}(\R) $ with a solution-map that is analytic from $H^{-1}(\R) $ to $C([0,T];H^{-1}(\R))$ whereas it is ill-posed in $ H^s(\R) $, as soon as $ s<-1 $, in the sense that the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\R) $ to even ${\cal D}'(\R) $ at any fixed $ t>0 $ small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be very us…
We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2019), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside of $H^\frac12(\mathbb{T})$. Here, we employ the method introduced by Deng-Nahmod-Yue (2019) to prove local well-posedness of the second renormalized mKdV equation in the Fourier-Lebesgue spaces $\mathcal{F}L^{s,p}(\mathbb{T})$ for $s\geq \frac12$ and $1\leq p <\infty$. As …
We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M^{1/4}_{2,q}(\mathbb{R})$ $(2\leq q\leq\infty)$. Moreover, we show that the data-to-solution map fails to be $C^3$ continuous in $M^{s}_{2,q}(\mathbb{R})$ when $s<1/4$. It is well-known that $H^{1/4}$ is a critical Sobolev space of mKdV so that it is well-posedness in $H^s$ for $s\geq 1/4$ and ill-posed (in the sense of uniform continuity) in $H^{s'}$ with $s'<1/4$. Noticing that $M^{1/4}_{2,q} \subset B^{1/q-1/4}_{2,q}$ is a sharp embedding and $H^{-1/4}\…
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.
We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle. We first consider the real-valued setting and show global well-posedness of the (usual) renormalized mKdV equation in the Fourier-Lebesgue spaces. In the complex-valued setting, we observe that the momentum plays an important role in the well-posedness theory. In particular, we prove that the complex-valued mKdV equation is ill-posed in the sense of non-existence of solutions when the momentum is infinite, in the spirit of the work on the nonlinear Schrödinger equation by Guo-Oh (2018). This non-existence res…
This paper is a continuation of the paper \emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}. In this paper, we consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as following: \begin{equation*} \begin{cases} \partial_t u - \partial_x^5 u + 30u^2\partial_x u + 20 u\partial_x u \partial_x^3u + 10u \partial_x^3 u = 0, \hspace{1em} (t,x) \in \mathbb{R} \times \mathbb{T}, u(0,x) = u_0(x) \in H^s(\mathbb{T}) \end{cases}. \end{equation*} We prove the local well-posedness of the fifth-order KdV equation for low regularity Sobolev in…
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
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.
We prove an endpoint multilinear estimate for the $X^{s,b}$ spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm. In particular we prove a multilinear estimate which completes the proof of global well-posedness for periodic KdV in a preceding paper (math.AP/0110045) down to the optimal regularity H^{-1/2}.
We consider the local well-posedness problem of a one-parameter family of coupled KdV-type systems both in the periodic and non-periodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter αwhen α\ne 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish sharp local well-posedness of the system in H^s(\mathbb{T}_λ), s \geq s^\ast, where s^\ast = s^\ast(α) \in ({1/2}, 1] is determined by the Diophantine characterization of certain constants derived from the coupling parameter α…
We prove special decay properties of solutions to the initial value problem associated to the $k$-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique continuation properties of solutions to this equation. As application of our method we also obtain results concerning the decay behavior of perturbations of the traveling wave solutions as well as results for solutions corresponding to special data.
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…
In this paper, we show the global well-posedness for periodic gKdV equations in the space $H^s(\mathbb{T})$, $s\ge \frac12$ for quartic case, and $s> \frac59$ for quintic case. These improve the previous results of I-team in 2004. In particular, the result is sharp for quintic case. The main approaches are the I-method combining with the resonance decomposition developed by Miao et al in 2010, and a bilinear Strichartz estimate in periodic setting.
Given smooth step-like initial data $V(0,x)$ on the real line, we show that the Korteweg--de Vries equation is globally well-posed for initial data $u(0,x) \in V(0,x) + H^{-1}(\mathbb{R})$. The proof uses our general well-posedness result for exotic spatial asymptotics. As a prerequisite, we show that KdV is globally well-posed for $H^3(\mathbb{R})$ perturbations of step-like initial data. In the case $V \equiv 0$, we obtain a new proof of the Bona--Smith theorem using the low-regularity methods that established the sharp well-posedness of KdV in $H^{-1}$.
The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for $H^s$ initial data, $s>-1/2$, and for any $s_1<\min(3s+1,s+1)$, the difference of the nonlinear and linear evolutions is in $H^{s_1}$ for all times, with at most polynomially growing $H^{s_1}$ norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case $s\geq 0$. Our result and a theorem of Oskolkov for the Airy evolution imply that if one starts with continuous and bounded variation initial data then the solution of KdV (given by the $L^2$ theory o…
Stationary solutions on a bounded interval for an initial-boundary value problem to Korteweg--de~Vries and modified Korteweg--de~Vries equation (for the last one both in focusing and defocusing cases) are constructed. The method of the study is based on the theory of conservative systems with one degree of freedom. The obtained solutions turn out to be periodic. Exact relations between the length of the interval and coefficients of the equations which are necessary and sufficient for existence of nontrivial solutions are established.
We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2π\mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.
Periodic waves of the modified Korteweg-de Vries (mKdV) equation are identified in the context of a new variational problem with two constraints. The advantage of this variational problem is that its non-degenerate local minimizers are stable in the time evolution of the mKdV equation, whereas the saddle points are unstable. We explore the analytical representation of periodic waves given by Jacobi elliptic functions and compute numerically critical points of the constrained variational problem. A broken pitchfork bifurcation of three smooth solution families is found. Two families represent (…
We consider the initial value problem of the fifth order modified KdV equation on the Sobolev spaces. \partial_t u - \partial_x^5u + c_1\partial_x^3(u^3) + c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0, u(x,0)= u_0(x) where $ u:R\timesR \to R $ and $c_j$'s are real. We show the local well-posedness in H^s(R) for s \geq 3/4 via the contraction principle on $X^{s,b}$ space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below $H^{3/4}(R)$. The counter example is obtained by approximating the fifth order mKdV equation by the cubic NLS equ…
In this article, we will prove $L^2(\mathbb{R})$-stability of $1$-solitons for the KdV equation by using exponential stability property of the semigroup generated by the linearized operator. The proof follows the lines of recent stability argument of Mizumachi [Asymptotic stability of lattice solitons in the energy space, Comm. Math. Phys., (2009)] and Mizumachi, Pego and Quintero [Asymptotic stability of solitary waves in the Benney-Luke model of water waves, Differential Integral Equations, (2013)] which show stability in the energy class by using strong linear stability of solitary waves in…
We prove global well-posedness of the Korteweg--de Vries equation for initial data in the space $H^{-1}(R)$. This is sharp in the class of $H^{s}(R)$ spaces. Even local well-posedness was previously unknown for $s<-3/4$. The proof is based on the introduction of a new method of general applicability for the study of low-regularity well-posedness for integrable PDE, informed by the existence of commuting flows. In particular, as we will show, completely parallel arguments give a new proof of global well-posedness for KdV with periodic $H^{-1}$ data, shown previously by Kappeler and Topalov, as …
We prove a first stability result of self-similar blow-up for the modified KdV equation on the line. More precisely, given a self-similar solution and a sufficiently small regular profile, there is a unique global solution which behaves at t tends to 0 as the sum of the self-similar solution and the smooth perturbation.
Given a suitable solution $V(t,x)$ to the Korteweg--de Vries equation on the real line, we prove global well-posedness for initial data $u(0,x) \in V(0,x) + H^{-1}(\mathbb{R})$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles $V(0,x) \in H^5(\mathbb{R}/\mathbb{Z})$ satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In our companion paper we show that smooth step-like initial data also satisfy our hypotheses. …
We study the long-time dynamics of complex-valued modified Korteweg-de Vries (mKdV) solitons, which are recognized because they blow-up in finite time. We establish stability properties at the H^1 level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are H^1 stable, improving our previous result, where we only proved H^2 stability. The main new ingredient of the proof is the use of a Bäcklund transformation which links the behavior of breathers, complex-valued solitons and small real-valued solutions of the mKdV equation. We als…
The Kawahara and modified Kawahara equations are fifth-order KdV type equations and have been derived to model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for Kawahara equation in $H^s({\mathbf R})$ with $s>-\frac74$ and the local well-posedness for the modified Kawahara equation in $H^s({\mathbf R})$ with $s\ge-\frac14$. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the $[k; Z]$ multiplier norm method o…
In this paper we obtain global well-posedness in low order Sobolev spaces of higher order KdV type equations with dissipation. The result is optimal in the sense that the flow-map is not twice continuously differentiable in rougher spaces. The solution is shown to be smooth for positive times.
