Related papers: The Korteweg-de Vries Equation on general star gra…
We prove local well-posedness for the Cauchy problem associated to Korteweg-de Vries equation on a metric star graph with three semi-infinite edges given by one negative half-line and two positives half-lines attached to a common vertex,…
This paper investigates a boundary-value problem for the Korteweg-de Vries (KdV) equation on a star-graph structure. We develop a unified framework introducing the notion of $s$-compatibility, which generalizes classical compatibility…
In this paper, we establish local well-posedness for the Cauchy problem associated with the Kawahara equation on a general metric star graph. Initially, we identify suitable boundary conditions that produce a well-behaved dynamics for the…
We consider the linearized Korteweg-de-Vries equa- tions, sometimes called Airy equation, on general metric graphs with edge lengths bounded away from zero. We show that pro- perties of the induced dynamics can be obtained by studying…
It is well known that the controllability property of partial differential equations (PDEs) is closely linked to the proof of an observability inequality for the adjoint system, which, sometimes, involves analyzing a spectral problem…
The aim of this work is to establish a linear instability criterium of stationary solutions for the Korteweg-de Vries model on a star graph with a structure represented by a finite collections of semi-infinite edges. By considering a…
In a recent article \textit{"Lower regularity solutions of the biharmonic Schr\"odinger equation in a quarter plane", to appear on Pacific Journal of Mathematics [15]}, the authors gave a starting point of the study on a series of problems…
We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k\geq5$ is an integer number and $\mu=\pm1$. In the focusing case ($\mu=1$), we show that if the initial data $u_0$…
The initial-boundary value problem for the generalized Korteweg-de Vries equation on a half-line is studied by adapting the initial value techniques developed by Kenig, Ponce and Vega and Bourgain to the initial-boundary setting. The…
We study the small amplitude linearization of the Korteweg de Vries equation on the line with a local defect scattering waves represented by a metric graph domain adjoined at one point. For a representative collection of examples, we derive…
We study the Korteweg--de Vries equation on a metric star graph and investigate existence of solitary waves on the metric graph in terms of the coefficients of the equation on each edge, the coupling condition at the central vertex of the…
Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects…
The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper…
In the present paper the Airy operator on star graphs is defined and studied. The Airy operator is a third order differential operator arising in different contexts, but our main concern is related to its role as the linear part of the…
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…
This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg-de Vries equation, the Kuramoto-Sivashinsky equation, the generalized Korteweg-de Vries-Kuramoto-Sivashinski equation and the non…
The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., $\psi^{p-2}\psi$) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local…
We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions \begin{align} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \nonumber &q(x,0)=q_{0}(x)\to \pm 1, \ \…
The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of…
Results on well-posedness of three inverse problems with integral conditions on a bounded interval for the generalized Korteweg-de Vries equation without any restrictions on the growth rate of nonlinearity are established. Either the…