Related papers: On the periodic Korteweg-de Vries equation: a norm…
Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other…
We prove two new mixed sharp bilinear estimates of Schr\"odinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schr\"odinger - Kortweg-deVries (NLS-KdV) system in the \emph{periodic setting}. Our…
The work presented here emanates from questions arising from experimental observations of the propagation of surface water waves. The experiments in question featured a periodically moving wavemaker located at one end of a flume that…
It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry…
We discuss a new numerical schema for solving the initial value problem for the Korteweg-de Vries equation for large times. Our approach is based upon the Inverse Scattering Transform that reduces the problem to calculating the reflection…
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…
We study soliton solutions to a generalized Korteweg - de Vries (KdV) equation with a saturated nonlinearity, following the line of inquiry of the authors for the nonlinear Schr\"odinger equation (NLS). KdV with such a nonlinearity is known…
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…
The periodic KdV equation u_t=u_{xxx}+\beta uu_x arises from a Hamiltonian system with infinite-dimensional phase space L^2(T). Bourgain has shown that there exists a Gibbs measure \nu on balls \{\phi :\Vert\Phi\Vert^2_{L^2}\leq N\} in the…
For the Schr\"odinger equation $u_t+i u_{xx}=\nab^\be[u^2]$, $\be\in (0,1/2)$, we establish local well-posedness in $H^{\be-1+}$ (note that if $\be=0$, this matches, up to an endpoint, the sharp result of Bejenaru-Tao, \cite{BT}). Our…
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\e^2$, $\e\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by…
We study the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with step-like initial data approaching nonzero constants $c_l$ and $c_r$ as $x \to -\infty$ and $x\to+\infty$, respectively. Assuming $c_l>c_r>0$,…
We consider the real-valued defocusing modified Korteweg-de Vries equation (mKdV) on the circle. Based on the complete integrability of mKdV, Killip-Vi\c{s}an-Zhang (2018) discovered a conserved quantity which they used to prove low…
In this paper, we study Lipschitz continuity of the solution mappings of regularized least-squares problems for which the convex regularizers have (Fenchel) conjugates that are $\mathcal{C}^2$-cone reducible. Our approach, by using…
In this paper we review the physical relevance of a Korteweg-de Vries (KdV) equation with higher-order dispersion terms which is used in the applied sciences and engineering. We also present exact traveling wave solutions to this…
The lattice potential Korteweg-de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated Korteweg-de Vries equation. These include…
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study dispersive equations with a time non-homogeneous modulation acting on the…
In this paper we establish the persistence property for solutions of the quartic generalized Korteweg-de Vries equation with initial data in weighted Sobolev spaces $H^{s}(\mathbb{R})\cap L^2(|x|^{2r}dx)$ for $s =1/12 + \varepsilon$ and any…
We investigate the points of constancy in the piecewise constant solution profiles of the periodic linearized Korteweg--deVries equation with step function initial data at rational times. The solution formulas are given by certain Weyl…
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…