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Alexei Rybkin

Publications and source records attributed to Alexei Rybkin.

At least 19 recordsLinked to original sources
arXiv2026

On the inverse scattering transform for the KdV equation with summable initial data

We consider the Cauchy problem for the Korteweg--de Vries equation with real initial data $q$ that is both $L^1$ and $L^2$ summable and supported on (0,\infty). Using the left reflection coefficient and Hankel operators on the Hardy space $H^2$, we derive a trace-type representation for the corresponding solution. The proof is based on approximation by compactly supported potentials, uniform convergence of the associated reflection coefficients away from the origin, and continuity properties of the resulting Hankel operators. This yields a rigorous inverse scattering construction for a class of summable half-line supported initial data beyond the standard short-range setting.

math-ph↗
arXiv2025

Continuous binary Darboux transformation as an abstract framework for KdV soliton gases

We present a unified operator-theoretic framework for constructing deterministic KdV soliton gases and step-type KdV solutions. Starting from Dyson's determinantal formula, we obtain a broad class of reflectionless solutions and describe their basic spectral and analytic properties, including their interpretation as deterministic soliton gases. We then introduce a continuous binary Darboux transformation that acts directly on the scattering data and generates general step-type solutions, with particular emphasis on reflectionless hydraulic-jump-type profiles modelling a soliton condensate on the left and vacuum on the right. The paper is methodological in nature: our goal is not to develop a full kinetic or probabilistic theory, but to show how classical tools from spectral and scattering theory can be combined into a conceptually simple framework that accommodates both reflectionless and non-reflectionless soliton gas configurations, including step-like backgrounds.

math-ph↗
arXiv2025

Reconstruction of the non-linear wave at a buoy from shoreline data and applications to the tsunami inverse problem for piece-wise sloping bathymetry

We discuss the following inverse problem: given the run-up data of a tsunami wave, can we recover its initial shape? We study this problem within the framework of the non-linear shallow water equations, a model widely used to study tsunami propagation and inundation. Previously, it has been demonstrated that in the case of infinite sloping bathymetry, it is possible to recover the initial water displacement and velocity from shoreline readings \cite{Rybkin23,Rybkin24,Rybkin25}. We consider a finite sloping bathymerty. We show that it is possible to recover boundary conditions (water displacement and velocity) on a virtual buoy from the shoreline data. Further, we discuss stitching together the shallow water equations and the Boussinesq equation in a more complex piece-wise sloping bathymetry in order to recover the initial conditions, while incorporating the dispersion to our model.

math.AP↗
arXiv2025

A new asymptotic regime for the KdV equation with Wigner-von Neumann type initial data

We investigate the long-time asymptotic behavior of solutions to the Cauchy problem for the KdV equation, focusing on the evolution of the radiant wave associated with a Wigner-von Neumann (WvN) resonance induced by the initial data (potential). A WvN resonance refers to an energy level where the potential exhibits zero transmission (complete reflection). The corresponding Jost solution at such energy becomes singular, and in the NLS context, this is referred to as a spectral singularity. A WvN resonance represents a long-range phenomenon, often introducing significant challenges, such as an infinite negative spectrum, when employing the inverse scattering transform (IST). To avoid some of these issues, we consider a restricted class of initial data that generates a WvN resonance but for which the IST framework can be suitably adapted. For this class of potentials, we demonstrate that each WvN resonance produces a distinct asymptotic regime -- termed the resonance regime -- characterized by a slower decay rate for large time compared to the radiant waves associated with short-range initial data.

math.AP↗
arXiv2025

Separation of the initial conditions in the inverse problem for 1D non-linear tsunami wave run-up theory

We investigate the inverse tsunami wave problem within the framework of the 1D nonlinear shallow water equations (SWE). Specifically, we focus on determining the initial displacement $\eta_0(x)$ and velocity $u_0(x)$ of the wave, given the known motion of the shoreline $R(t)$ (the wet/dry free boundary). We demonstrate that for power-shaped inclined bathymetries, this problem admits a complete solution for any $\eta_0$ and $u_0$, provided the wave does not break. In particular, we show that the knowledge of $R(t)$ enables the unique recovery of both $\eta_0(x$) and $u_0(x)$ in terms of the Abel transform. It is important to note that, in contrast to the direct problem (also known as the tsunami wave run-up problem), where $R(t)$ can be computed exactly only for $u_0(x)=0$, our algorithm can recover $\eta_0$ and $u_0$ exactly for any non-zero $u_0$. This highlights an interesting asymmetry between the direct and inverse problems. Our results extend the work presented in \cite{Rybkin23,Rybkin24}, where the inverse problem was solved for $u_0(x)=0$. As in previous work, our approach utilizes the Carrier-Greenspan transformation, which linearizes the SWE for inclined bathymetries. Extensive numerical experiments confirm the efficiency of our algorithms.

math.AP↗
arXiv2023

Inverse non-linear problem of the long wave run-up on coast

The study of the process of catastrophic tsunami-type waves on the coast makes it possible to determine the destructive force of waves on the coast. In hydrodynamics, the one-dimensional theory of the run-up of non-linear waves on a flat slope has gained great popularity, within which rigorous analytical results have been obtained in the class of non-breaking waves. In general, the result depends on the characteristics of the wave approaching (or generated on) the slope, which is usually not known in the measurements. Here we describe a rigorous method for recovering the initial displacement in a source localised in an inclined power-shaped channel from the characteristics of a moving shoreline. The method uses the generalised Carrier-Greenspan transformation, which allows one-dimensional non-linear shallow-water equations to be reduced to linear ones. The solution is found in terms of Erd\'elyi-Kober integral operator. Numerical verification of our results is presented for the cases of a parabolic bay and an infinite plane beach.

physics.flu-dyn↗
arXiv2023

Trace formulas revisited and a new representation of KdV solutions with short-range initial data

We put forward a new approach to Deift-Trubowitz type trace formulas for the 1D Schrodinger operator with potentials that are summable with the first moment (short-range potentials). We prove that these formulas are preserved under the KdV flow whereas the class of short-range potentials is not. Finally, we show that our formulas are well-suited to study the dispersive smoothing effect.

math-ph↗
arXiv2022

A continuous analog of the binary Darboux transformation for the Korteweg-de Vries equation

In the KdV context we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann-Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step-type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context our method offers same benefits as the classical binary Darboux transformation does.

math-ph↗
arXiv2021

Norming constants of embedded bound states and bounded positon solutions of the Korteweg-de Vries equation

In the context of the full line Schrodinger equation, we revisit the binary Darboux transformation (double commutation method) which inserts or removes any number of positive eigenvalues embedded into the absolutely continuous spectrum without altering the rest of scattering data. We then show that embedded eigenvalues produce an additional explicit term in the KdV solution. This term looks similar to multi-soliton solution and describes waves traveling in the direction opposite to solitons. It also resembles the known formula for (singular) multi-positon solutions but remains bounded, which answers in the affirmative Matveev's question about existence of bounded positons.

math-ph↗
arXiv2020

General Initial Value Problem for the Nonlinear Shallow Water Equations: Runup of Long Waves on Sloping Beaches and Bays

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier-Greenspan transformation [G. Carrier and H. Greenspan, J. Fluid Mech. 01, 97 (1957)]. We use a Taylor series approximation to deal with the difficulty associated with the initial conditions given on a curve in the transformed space. This extends earlier solutions to waves with near shore initial conditions, large initial velocities, and in more complex U-shaped bathymetries; and allows verification of tsunami wave inundation models in a more realistic 2-D setting.

physics.flu-dyn↗
arXiv2019

The effect of a positive bound state on the KdV solution. A case study

We consider a slowly decaying oscillatory potential such that the corresponding 1D Schr\"odinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial data for which the Cauchy problem for the Korteweg-de Vries (KdV) equation can be solved by the inverse scattering transform. We nevertheless show that the KdV equation with our potential does admit a closed form classical solution in terms of Hankel operators. Comparing with rapidly decaying initial data our solution gains a new term responsible for the positive eigenvalue. To some extend this term resembles a positon (singular) solution but remains bounded. Our approach is based upon certain limiting arguments and techniques of Hankel operators.

math-ph↗
arXiv2019

Method for solving hyperbolic systems with initial data on non-characteristic manifolds with applications to the shallow water wave equations

We are concerned with hyperbolic systems of order-one linear PDEs originated on non-characteristic manifolds. We put forward a simple but effective method of transforming such initial conditions to standard initial conditions (i.e. when the solution is specified at an initial moment of time). We then show how our method applies in fluid mechanics. More specifically, we present a complete solution to the problem of long waves run-up in inclined bays of arbitrary shape with nonzero initial velocity.

math.AP↗
arXiv2019

On classical solutions of the KdV equation

\begin{abstract} We show that if the initial profile $q\left( x\right) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $\int^{\infty }x^{5/2}\left\vert q\left( x\right) \right\vert dx<\infty,$ (no decay at $-\infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. \end{abstract}

math-ph↗
arXiv2017

KdV equation beyond standard assumptions on initial data

We show that the Cauchy problem for the KdV equation can be solved by the inverse scattering transform (IST) for any initial data bounded from below, decaying sufficiently rapidly at plus infinity, but unrestricted otherwise. Thus our approach doesn't require any boundary condition at minus infinity.

math-ph↗
arXiv2013

Soliton theory and Hankel operators

Soliton theory and the theory of Hankel (and Toeplitz) operators have stayed essentially hermetic to each other. This paper is concerned with linking together these two very active and extremely large theories. On the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation we demonstrate the power of the language of Hankel operators in which symbols are conveniently represented in terms of the scattering data for the Schrodinger operator associated with the initial data for the KdV equation. This approach yields short-cuts to already known results as well as to a variety of new ones (e.g. wellposedness beyond standard assumptions on the initial data) which are achieved by employing some subtle results for Hankel operators.

math.AP↗