On the Cauchy problem for the Korteweg–de Vries equation with steplike finite-gap initial data: I. Schwartz-type perturbations

Abstract

We solve the Cauchy problem for the Korteweg-de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finitegap potentials under the assumption that the respective spectral bands either coincide or are disjoint.

Rarefaction waves of the Korteweg–de Vries equation via nonlinear steepest descent

“…For recent developments and further information we refer to [9]. The application of the inverse scattering transform to the problem (1.1)-(1.2) (see [10], [11]) implies that the solution q(x, t) of the Cauchy problem exists in the classical sense and is unique in the class…”

Dispersive and soliton perturbations of finite-genus solutions of the KdV equation: Computational results

“…We have combined the dressing method and Riemann-Hilbert contour deformations with a numerical method for RHPs to compute a class of step-like finite-genus solutions of the KdV 7 equation [11] which we call superposition solutions. Due to either quasi-periodicity ( Figure 3) or the induced phase shift ( Figure 5) no other existing numerical methods can compute these solutions.…”

“…The corresponding numerical method was discussed in detail in [10]. The analysis of such superposition solutions was examined by Egorova, Grunert and Teschl [11] and by Mikikits-Leitner & Teschl [12], including some scenarios that are not covered in [10].…”

Asymptotic Multiphase Solitonlike Solutions of the Cauchy Problem for a Singularly Perturbed Korteweg–de-Vries Equation with Variable Coefficients

“…Note that the Cauchy problem for the Korteweg-de Vries equation and its generalizations were studied in numerous papers [10][11][12][13][14][15][16][17][18][19][20][21]; see also the survey [22]. Thus, in particular, the problem of existence of solutions of the Cauchy problem for the Korteweg-de Vries equation with variable coefficients was studied in [19][20][21].…”