Alexei Rybkin

• University of Alaska Fairbanks

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 refle...

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 incline...

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 n...

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...

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...

In the Korteweg–de Vries equation (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 r...

We show that if the initial profile \(q\left ( x\right ) \) for the Korteweg-de Vries (KdV) equation is supported on \(\left ( a,\infty \right ) ,a>-\infty ,\) and \(\int _{a}^{\infty }x^{7/4}\left \vert q\left ( x\right ) \right \vert dx<\infty ,\) then the time evolved \(q\left ( x,t\right ) \) is a classical solution of the KdV equation.

We put forward a simple but effective explicit method of recovering initial data for the nonlinear shallow water system from the reading at the shoreline. We then apply our method to the tsunami waves inverse problem.

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 explici...

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. Th...

In the context of the Cauchy problem for the Korteweg–de Vries equation we extend the inverse scattering transform to initial data that behave at plus infinity like a sum of Wigner–von Neumann type potentials with small coupling constants. Our arguments are based on the theory of Hankel operators.

In the context of the Cauchy problem for the Korteweg-de Vries equation we extend the inverse scattering transform to initial data that behave at plus infinity like a sum of Wigner-von Neumann type potentials with small coupling constants. Our arguments are based on the theory of Hankel operators.

CANWA (Comparative Analytical Numerical Wave Algorithm), written in MATLAB, provides a fast, direct comparison of a general finite volume solution to the 1+1 shallow water wave equations with a robust analytical solution recently presented by Nicolsky et al. (2018). The implementation of the method for data projection, introduced in Nicolsky et al....

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 explici...

We present an exact analytical solution for computations of runup in constantly inclined U- and V-shaped bays. The provided solution avoids integration of indefinite double integrals in (Rybkin et al., Water Waves 3(1):267–296, 2021) and is based on a simple analytic expression for the Green’s function. We analyze wave runup in parabolic and certai...

In the KdV context, we revisit the classical Darboux transformation in the framework of the vector Riemann–Hilbert problem. This readily yields a version of the binary Darboux transformation providing a short‐cut to explicit formulas for solitons traveling on a wide range of background solutions. Our approach also links the binary Darboux transform...

We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross section by means of the generalized Carrier–Greenspan hodograph transform (Rybkin et al. in J Fluid Mech, 748:416–432, 2014). Since the Carrier–Greenspan transform, while linearizing the shallow water...

We consider a slowly decaying oscillatory potential such that the corresponding 1D Schrödinger 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 scatter...

We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross-section by means of the generalized Carrier-Greenspan hodograph transform (Rybkin et al., 2014). Since the Carrier-Greenspan transform, while linearizing the shallow water system, seriously entangles t...

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 transf...

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....

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 scatt...

\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....

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....

Мы изучаем свойство принадлежности оператора Ганкеля (и его производных по параметру) с сильно осциллирующим символом классу ядерных операторов. Наш подход базируется на критерии Пеллера о ядерности операторов Ханкеля и точном анализе, возникающего при этом тройного интеграла с помощью метода перевала. Полученные результаты представляются оптимальн...

The trace-class property of Hankel operators (and their derivatives with respect to the parameter) with strongly oscillating symbol is studied. The approach used is based on Peller’s criterion for the trace-class property of Hankel operators and on the precise analysis of the arising triple integral using the saddle-point method. Apparently, the ob...

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier–Greenspan transformation (Carrier and Greenspan (1957) [9]). 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...

Modeling of tsunamis in glacial fjords prompts us to evaluate applicability of the cross-sectionally averaged nonlinear shallow water equations to model propagation and runup of long waves in asymmetrical bays and also in fjords with two heads. We utilize the Tuck-Hwang transformation, initially introduced for the plane beaches and currently genera...

In the context of the Cauchy problem for the Korteweg-de Vries equation we put forward a new effective method to link smoothness of the solution with the rate of decay of the initial data. Our approach is based on the Peller characterization of trace class Hankel operators.

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.

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.

We present an analytical study of the propagation and run-up of long waves in piecewise sloping, U-shaped bays using the cross-sectionally averaged shallow water equations. The nonlinear equations are transformed into a linear equation by utilizing the generalized Carrier–Greenspan transform (Rybkin et al. J Fluid Mech 748:416–432, 2014). The solut...

Run-up of long waves in sloping U-shaped bays is studied analytically in the framework of the 1-D nonlinear shallow water theory. By assuming that the wave flow is uniform along the cross section, the 2-D nonlinear shallow water equations are reduced to a linear semi-axis variable-coefficient 1-D wave equation via the generalized Carrier-Greenspan...

By assuming the flow is uniform along the narrow long bays, the 2-D nonlinear shallow-water equations are reduced to a linear semi-axis variable-coefficient 1-D wave equation via the generalized Carrier–Greenspan transformation. The run-up of long waves in constantly sloping U-shaped and V-shaped bays is studied both analytically and numerically wi...

Long nonlinear wave runup on the coasts of trapezoidal bays is studied analytically in the framework of one-dimensional (1-D) nonlinear shallow-water theory with cross-section averaging, and is also studied numerically within a two-dimensional (2-D) nonlinear shallow water theory. In the 1-D theory, it is assumed that the trapezoidal cross-section...

We develop the inverse scattering transform for the KdV equation with real singular initial data $q(x)$ of the form $q(x) = r'(x) + r(x)^2$, where $r\in L^2_{\textrm{loc}}$ and $r=0$ on $\mathbb R_+$. As a consequence we show that the solution $q(x,t)$ is a meromorphic function with no real poles for any $t>0$.

We show that the KdV flow evolves any real locally integrable initial profile q of the form q = r′+ r2, where r ∈ L2loc, r|ℝ+ = 0 into a meromorphic function with no real poles.

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 Hank...

We derive an asymptotic formula for the argument of a Blaschke product in the upper half-plane with purely imaginary zeros. We then use this formula to find conditions for the quotient of two such Blaschke products to be continuous on the real line. These results are applied to certain Hankel and Toeplitz operators arising in the Cauchy problem for...

In this work we present a rigorous analytical solution of nonlinear shallow water theory for wave runup in inclined channels of arbitrary cross-section, which generalizes previous studies for wave runup on a plane beach and in channels of parabolic cross-section. The solution is found using a hodograph transformation, which generalizes the well-kno...

We discuss a numerical schema for solving the initial value problem for the Korteweg-de Vries equation in the soliton region which is based on a new method of evaluation of bound state data. Using a step-like approximation of the initial profile and a fragmentation principle for the scattering data, we obtain an explicit procedure for computing the...

We are concerned with the inverse scattering problem for the full line Schrödinger operator −∂²x + q(x) with a steplike potential q a priori known on . Assuming is known and short range, we show that the unknown part of q can be recovered by where is the classical Marchenko operator associated with and is a trace class integral Hankel operator. The...

In the context of the Korteweg-de Vries equation we put forward some new conservation laws which hold for real initial profiles with low regularity. Some applications to spectral theory of the one-dimensional Schrödinger operator with singular potentials are also considered. KeywordsKorteweg-de Vries equation–modified perturbation determinants–con...

We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles q's which are, in a certain sense, essentially bounded from below and q(x)=O(e^{-cx^{{\epsilon}}}),x\rightarrow+\infty, with some positive c and {\epsilon}. Using the inverse scattering transform, we show that the KdV flow turns such init...

We show that the KdV flow evolves any real singular initial profile q of the form q=r'+r^2, where r\inL_{loc}^2, r|_{R_+}=0 into a meromorphic function with no real poles.

We discuss a new numerical schema for solving the initial value problem for the Korteweg-de Vries equation for large times. Our approach is based upon the Inverse Scattering Transform that reduces the problem to calculating the reflection coefficient of the corresponding Schr\"odinger equation. Using a step-like approximation of the initial profile...

We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V0 which is decaying sufficiently fast at +∞ and arbitrarily enough (i.e. no decay or pattern of behaviour) at −∞. We show that this system is completely integrable in a very strong sense. Namely, the solution V(x, t) admits the Hirota τ-function...

We show that under the Korteweg–de Vries flow an initial profile supported on (−∞, 0) from a very broad function class (without any decay assumption) instantaneously evolves into a meromorphic function with no poles on the real line. Our treatment is based on a suitable modification of the inverse scattering transform and a detailed investigation o...

For the Titchmarsh-Weyl m-function of the half-line Schrödinger operator with Dirichlet boundary conditions we put forward a new interpolation formula which allows one to reconstruct the m-function from its values on a certain infinite set of points for a broad class of potentials.

For the Schrödinger operator H = −∂ 2 x + q(x) on L 2 (−∞, ∞) with a real potential q(x) from a broad class of non-decaying functions, we extend the well-known Marchenko inverse scattering method to recover q(x) from a suitably defined reflection coefficient R and the knowledge of q(x) on (0, ∞). Our main tool is the Titchmarsh–Weyl m-function asso...

We link boundary control theory and inverse spectral theory for the Schrödinger operator H = @2 x+q (x) on L2 (0;1) with Dirichlet boundary condition at x = 0: This provides a shortcut to some results on inverse spectral theory due to Simon, Gesztesy-Simon and Remling. The approach also has a clear physical interpritation in terms of boundary contr...

We are concerned with the Korteweg–de Vries equation on the full line with real nondecaying initial profiles. We find the time evolution of a (relative) reflection coefficient. An inverse spectral formalism is also considered for certain mixed problems on the full line.

Let T be a contraction that is a trace class perturbation of a unitary operator V, and let {λ k } be the discrete spectrum of T. For a sufficiently large class of functions Φ the trace formula tr{Φ(T)-Φ(V)}=∑ k {Φ(λ k )-Φ(λ k /|λ k |)}+(B)∫ 0 2π Φ ' (e iφ )dΩ(φ) holds. This formula is a direct analogue of the well-known M. G. Kreĭn trace formula fo...

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the A−amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl m−function associated with the Schrödinger operator H = −∂ x 2 + q(x) on L 2(0, ∞) with Dirichlet boundary condition at x = 0.

We prove that the reflection coefficient of one-dimensional Schrödinger operators with potentials supported on a half-line can be represented in the upper half-plane as the quotient of a contractive analytic function and a properly regularized Blaschke product. We apply this fact to obtain a new trace formula and trace inequality for the reflection...

Let H = −∂x2+V(x) be a properly defined Schrödinger operator on L2() with real potentials of the form V(x) = q(x)+p′(x) (the derivative is understood in the distributional sense) with some p,q ∊ L2(). We prove that the absolutely continuous spectrum of H fills (0,∞) which was previously proven by Deift-Killip for V ∊ L2(). We also refine the 3/2-Li...

For one-dimensional Schrödinger operators with potentials q subject to ∑n = −∞∞( ∫ nn+1∣q(x)∣dx)2<∞, we prove that the absolutely continuous spectrum is [0,∞), extending the 1999 result due to Dieft–Killip. As a by-product we show that under the same condition the sequence of the negative eigenvalues is 3/2-summable improving the relevant result by...

We put forward a new transformation of the half-line Sturm-Liouville equation with non-smooth potentials from L p ,with p ≥ 2. This transformation yields existence of the Weyl solution with higher order WKB-type asymptotic behavior (spatial and spectral parameter). We apply our approach to the study of high-energy asymptotics for the Titchmarsh-Wey...

We deal with trace formulas for half-line Schrödinger operators with long-range potentials. We generalize the Buslaev–Faddeev trace formulas to the case of square integrable potentials. The exact relation between the number of the trace formulas and the number of integrable derivatives of the potential is also given. The main results are optimal. ©...

For the general one dimensional Schrödinger operator -d²/dx²+q(x) with real q we study some analytic aspects related to order-one trace formulas originally due to Buslaev-Faddeev, Faddeev-Zakharov, and Gesztesy-Holden-Simon-Zhao. We show that the condition q∈ L₁( R) guarantees the existence of the trace formulas of...

Let J be a Jacobi real symmetric matrix on l 2 with zero diagonal and non-diagonal entries of the form f1 + png. If pn 1 pn = O(n ) with some > 2=3, then we prove the existance of bounded solutions of Ju = u for a.e. 2 ( 2; 2) with the WKB-type asymptotic behavior. 1.

For the general one-dimensional Schrödinger operator −d2/dx2+q(x) with real q [set membership] L1([open face R]), this paper presents a new series representation of the Jost solution which, in turn, implies a new asymptotic representation of the Weyl m-function for locally summable q. This representation is then applied to smooth potentials q to...

For the general one dimensional Schrödinger operator -d2/dx2 + q (x) with real q ∈ L1 (R) we present a complete streamlined treatment of large spectral parameter power asymptotics of Jost solutions and the scattering matrix. We find simple necessary and sufficient conditions relating the number of exact terms in the asymptotics with the smoothness...

The author presents an elementary procedure to derive certain trace relations for Schrödinger operators in one dimension relating potentials with some scattering data. In this way, he obtains some new trace type formulas as well as known ones previously studied by Gesztesy, Holden, Simon, and others, his a priori hypothesis on potentials being mini...

A new approach to deriving trace-type formulas is given. In this way, a new representation for the densities of the first integrals of the Korteweg-de Vries equation are found in terms of spectral and scattering data of the associated Schrödinger operator. We also utilize our method to improve many already-known results on the KdV invariants and as...

We put forward a direct analog of the Cauchy formula for analytic functions f in the unit disk from Hardy classes H p with 1/2≤p>1. In this way, one has to consider a weakened Stieltjes type integration with respect to radial means of f which are in general not measures. A suitable concept of integration is found which also lets us obtain a general...

We propose an approach to obtaining new trace formulas of the Gel’fand–Levitan–Buslaev–Faddeev type, valid for Hilbert–Schmidt perturbations. In this way we obtain a new trace formula for Schrödinger operators on the half-line with long-range potentials. © 1999 American Institute of Physics.

In terms of the characteristic function of the contraction the norms of the spectral projections onto its absolutely continuous component are evaluated.The row of examples is considered. The exposition is conducted in the framework of the Sz.-Nagy-Foias model.

It is shown that the naturally defined argument of a Blaschke product is a function which is the harmonic conjugate of an integrable function of constant sign. A direct construction of this function is obtained. This fact allows us to investigate the relation between conditions on the zeros of a Blaschke product and the convergence of the arguments...

In terms of the characteristic function of the contraction the norms of the spectral projections onto its absolutely continuous component are evaluated. The row of examples is considered. The exposition is conducted in the framework of the Sz.-Nagy-Foias model.

A trace formula is given for an abstract pair consisting of a dissipative operator and a selfadjoint operator, and a connection is established between the spectral shift function of this pair and the corresponding scattering matrix. As a consequence, trace formulas are obtained for a specific dissipative operator arising in the problem of resonance...

We are concerend with the KdV equation on the full-line with real non-decaying initial pro…le. We …nd the time evolution of a (relative) re‡ection coe¢ cient. An inverse spectral formalism is also considered for a certain mixed problems on the full-line.