Related papers: The generalized periodic ultradiscrete KdV equatio…
In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions…
We construct generalized solutions to the ultradiscrete KdV equation, including the so-called negative solition solutions. The method is based on the ultradiscretization of soliton solutions to the discrete KdV equation with gauge…
In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated. The full scope of obtaining…
We propose the algebro-geometric mothod of construction of solutions of the discrete KP equation over a finite field. We also perform the corresponding reduction to the finite field version of the discrete KdV equation. We write down…
Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by…
We consider matrices on infinite trees which are universal covers of Jacobi matrices on finite graphs. We are interested in the question of the existence of sequences of finite covers whose normalized eigenvalue counting measures converge…
Near linear evolution in Korteweg de Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction.
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…
An initial-boundary value problem for a generalized KdV equation posed on a half-line is considered. Existence and uniqueness of global regular solutions for arbitrary smooth initial data are established.
We study a 2-dimensional Box-Ball system which is a ultradiscrete analog of the discrete KP equation. We construct an algorithm to calculate the fundamental cycle, which is an important conserved quantity of the 2-dim. Box-Ball system with…
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…
We provide a framework to build periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary…
We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We…
We propose the functions defined by the maximum of a discrete quadratic form and satisfying the ultradiscrete KdV equation. These functions includes not only soliton solutions but also pseudo-periodic solutions. In the proof, we employ some…
Dispersive averaging effects are used to show that KdV equation with periodic boundary conditions possesses high frequency solutions which behave nearly linearly. Numerical simulations are presented which indicate high accuracy of this…
We establish a matrix generalization of the ultradiscrete fourth Painlev\'e equation (ud-PIV). Well-defined multicomponent systems that permit ultradiscretization are obtained using an approach that relies on a group defined by constraints…
We analyze the multisymplectic Preissman scheme for the KdV equation with the periodic boundary condition and show that the unconvergence of the widely-used iterative methods to solve the resulting nonlinear algebra system of the Preissman…
The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary…
We consider a class of first-order partial differential operators, acting on the space of ultradifferentiable periodic functions, and we describe their range by using the following conditions on the coefficients of the operators: the…
We investigate the multi-soliton solutions to the generalized discrete KdV equation. In some cases a soliton with smaller amplitude moves faster than that with larger amplitude unlike the soliton solutions of the KdV equation. This…