We derive a Riemann–Hilbert problem satisfied by the Baker-Akhiezer function for the finite-gap solutions of the Korteweg-de Vries (KdV) equation. As usual for Riemann-Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We make use of recent advances in the nume...

We introduce a deep neural network learning scheme to learn the B\"acklund transforms (BTs) of soliton evolution equations and an enhanced for data-driven equation discovery based on known BTs, respectively. The first takes advantage some solution (or equation) information study BT sine-Gordon equation, complex real Miura between defocusing (focusing) mKdV KdV as well via transforms. second use...

The Virasoro-Bott group endowed with the right-invariant L2metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

We revisit the symmetry structure of integrable PDEs, looking at specific example KdV equation. identify 4 nonlocal symmetries depending on a parameter, which we call generating symmetries. explain that since these are symmetries, their commutator algebra is not uniquely determined, and present three possibilities for algebra. In first version, 3 commute; this shows it possible to add further (...

In this paper, we find the relationship between the solution of (1+1)-dimensional Korteweg-de Vries (KdV) equation and the solution of (2+1)-dimensional integrable Schwarz-Korteweg-de Vries(SKdV) equation with Möbius transformations, Miura transformation and other transformations. Furthermore, we can obtain the new solution of the SKdV equation in 2+1 dimensions by using the solution of KdV equ...

Abstract We justify rigorously the convergence of the amplitude of solutions of Nonlinear-Schrödinger type Equations with non zero limit at infinity to an asymptotic regime governed by the Kortewegde Vries equation in dimension 1 and the Kadomtsev-Petviashvili I equation in dimensions 2 and more. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that t...