arXiv subjects

Ivan Toledo

Publications and source records attributed to Ivan Toledo.

4 recordsLinked to original sources

arXiv2026

The Marchenko method for soliton solutions to the Sawada--Kotera equation

Associated with the third-order linear differential operator, we present the Marchenko integral equation using as input the bound-state poles of a transmission coefficient and the time-evolved bound-state dependency constants. We derive the $\mathbf N$-soliton solution to the Sawada--Kotera equation, for an arbitrary positive integer $\mathbf N,$ by recovering that soliton solution from the solution to our Marchenko integral equation. Our method explains the origin of the $2\mathbf N$ real parameters appearing in the $\mathbf N$-soliton solution formula obtained by the ad-hoc method of Hirota. We show that $\mathbf N$ of those parameters are related to the $\mathbf N$ bound-state poles of the left transmission coefficient and the remaining $\mathbf N$ parameters are related to the bound-state dependency constants. Our Marchenko integral equation corresponds to the ``GLM (Gel'fand--Levitan--Marchenko) integral equation'' Kaup relentlessly but unsuccessfully tried to obtain.

nlin.SI↗

arXiv2025

Soliton solutions to the Sawada--Kotera equation

We consider the direct and inverse scattering problems for the third-order differential equation in the reflectionless case. We formulate a corresponding Riemann--Hilbert problem using input consisting of the bound-state poles of a transmission coefficient and the bound-state dependency constants. With the time-evolved dependency constants, using the solution to the Riemann--Hilbert problem, we construct soliton solutions to an integrable system of fifth-order nonlinear partial differential equations. By imposing some appropriate restrictions on the dependency constants, we show that those soliton solutions yield soliton solutions to the Sawada--Kotera equation.

nlin.SI↗

arXiv2025

Inverse scattering problem for the third-order equation on the line

We consider the third-order linear differential equation $$\displaystyle\frac{d^3\psi}{dx^3}+Q(x)\,\displaystyle\frac{d\psi}{dx}+P(x)\,\psi=k^3\,\psi,\qquad x\in\mathbb R,$$ where the complex-valued potentials $Q$ and $P$ are assumed to belong to the Schwartz class. We describe the basic solutions, the scattering coefficients, and the bound-state information, and we introduce the dependency constants and the normalization constants at the bound states. When the secondary reflection coefficients are zero, we provide a method to solve the corresponding inverse scattering problem, where the goal is to recover the two potentials $Q$ and $P$ from the scattering data set consisting of the transmission and primary reflection coefficients and the bound-state information. We formulate the corresponding inverse scattering problem as a Riemann--Hilbert problem on the complex $k$-plane and describe how the potentials are recovered from the solution to the Riemann--Hilbert problem. In the absence of bound states, we introduce a linear integral equation, which is the analog of the Marchenko integral equation used in the inverse scattering theory for the full-line Schr\"odinger equation. We describe the recovery of the two potentials from the solution to the aforementioned linear integral equation.

math-ph↗

arXiv2024

Soliton solutions associated with a class of third-order ordinary linear differential operators

Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation $d^3\psi/dx^3+Q\,d\psi/dx+P\psi =k^3\psi,$ where $Q$ and $P$ are the potentials in the Schwartz class and $k^3$ is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada--Kotera equation and the modified bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant $\mathbf N$-soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota.

nlin.SI↗