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 system and the generalized Miura transform to get weak continuity for the KdV equation.
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 $…
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 …
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 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…
We prove the existence of the global attractor in $ \dot H^s$, $s > 11/12$ for the weakly damped and forced mKdV on the one dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space is established. We directly apply the I-method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresp…
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 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}\…
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.
Generalized KdV equation subject to a stochastic perturbation
我们证明了次临界广义Korteweg-de Vries方程（mKdV和gKdV与四次幂的非线性）的全局适定性。更精确地说，我们证明了如果驱动噪声是$L^2(\mathbb{R})$上的圆柱Wiener过程，协方差算子是适当的索伯列夫空间中的Hilbert-Schmidt，那么具有$H^1(\mathbb{R})$数据的解在$H^1(\mathbb{R})$中是全局适定性的.这推广了A. de Bouard和A.德彪西的随机KdV方程。
On the fifth order KdV equation: local well-posedness and lack of uniform continuity of the solution map
本文证明了由KdV族$ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $产生的五阶方程在$ H^s(\mathbb{R}) $中是局部适定的，其中$ s> 5/2. Also, we prove the solution map of the equation is not uniformly continuous for $ s>0$.
Well-posedness and ill-posedness of the fifth order modifed KdV equation
在Sobolev空间上考虑五阶修正KdV方程的初值问题. \partial_t u = \partial_x^5 u + c_1 \partial_x^3（u^3）+\partial_x u + c_3uuPDMATHABMO u =0，u（x，0）= u_0（x）其中$ u:R\timesR \to R $和$c_j$是实数.利用$X^{s,b}$空间上的压缩原理证明了sPDMATHABMT 3/4在H^s（R）中的局部适定性.此外，我们表明，从数据到解决方案的解决方案映射未能一致连续低于$H^{3/4}(R)$。反例是通过用三次NLS方程近似五阶mKdV方程得到的。
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…
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 (…
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 …
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.
We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there …
We present a rigorous theory of a unified and simple inverse scattering transform (IST) for both focusing and defocusing real nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity. The IST problems for the nonlocal equations with NZBCs are more complicated then ones for the local equations with NZBCs. The suitable uniformization variable is introduced in order to make the direct and inverse problems be established on a complex plane instead of a two-sheeted Riemann surface. The direct scattering problem establishes the a…
In this paper, we first introduce a new function space $MH^{θ, p}$ whose norm is given by the $\ell^p$-sum of modulated $H^θ$-norms of a given function. In particular, when $θ< -\frac 12$, we show that the space $MH^{θ, p}$ agrees with the modulation space $M^{2, p}(\mathbb R)$ on the real line and the Fourier-Lebesgue space $\mathcal F L^{p}(\mathbb T)$ on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space setting. By applying the scaling symmetry, we then prove global well-pose…
Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Baecklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg-de Vries equations. Matrix equation can be viewed as a specialisation of operator equations in the finite dimensional case when operators are finite dimensional and, hence, admit a matrix representation. Baecklund transformations allow to reveal structural properties [S. Carillo and C. Schiebold, J. Math. Phys. 50 (2009), 073510] enjoyed by non-commut…
