Publications
A. Rybkin
Method for solving hyperbolic systems with initial data on
non-characteristic manifolds with applications to the shallow
water wave equations
Applied Mathematics Letters 93 (2019), 72–78.
S. Grudsky, A. Rybkin
On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg–de Vries Equation.
Mathematical Notes 104 (2018), no. 3, pp. 377–394.
D. Nicolsky, E. Pelinovsky, A. Raz, A. Rybkin
General initial value problem for the nonlinear shallow water equations: Runup of long waves on sloping beaches and bays.
Physics Letters A 382 (2018) issue 38, pp. 2738–2743
A. Rybkin
On Peller's Characterization of Trace Class Hankel Operators and Smoothness of KdV Solutions.
Proceedings of the American Mathematical Society 146 (2018), 1627-1637
A. Rybkin
KdV equation beyond standard assumptions on initial data.
Physica D: Nonlinear Phenomena 365 (2018), 1-11
D. Anderson, M. Harris, H. Hartle, D. Nicolsky, E. Pelinovsky, A. Raz, A. Rybkin
Run-Up of Long Waves in Piecewise Sloping U-Shaped Bays.
Pure and Applied Geophysics 174 (2017), Issue 8, pp. 3185–3207
A. Raz, D. Nicolsky, A. Rybkin, E. Pelinovsky
Long Wave Runup in Asymmetric Bays and in Fjords With Two Separate Heads.
Journal of Geophysical Research: Oceans 123 (2017) 2066-2080
V. Garayshin, M. Harris, D. Nicolsky, E. Pelinovsky, A. Rybkin
An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays.
Applied Mathematics and Computation 279 (2016) 187-197
M. Harris, D. Nicolsky, E. Pelinovsky, J. Pender, A. Rybkin
Run-up of nonlinear long waves in U-shaped bays of finite length: analytical theory and numerical computations.
Journal of Ocean Engineering and Marine Energy 2 (2016) 113-127
S. Grudsky, A. Rybkin
Soliton Theory and Hankel Operators.
SIAM Journal on Mathematical Analysis 47 (2015), no. 3, pp. 2283-2323
S. Grudsky, C. Remling, A. Rybkin.
The inverse scattering transform for the KdV equation with step-like singular Miura initial profiles.
Journal of Mathematical Physics 56 (2015), no. 9, 091505.
S. Grudsky, A. Rybkin.
Soliton theory and Hankel operators.
SIAM Journal of Mathematical Analysis 47 (2015), no. 3, pp. 2283-2323.
M. Harris, D. Nicolski, E. Pelinovsky, A. Rybkin.
Runup of nonlinear long waves in trapezoidal bays: 1-D analytical theory and 2-D numerical computations.
Pure and Applied Geophysics 172 (2015), 885-899.
A. Rybkin, E. Pelinovsky, I. Didenkulova.
Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier-Greenspan approach.
Journal of Fluid Mechanics 748 (2014), 416-432.
S. Grudsky, A. Rybkin.
On positive type initial profiles for the KdV equation.
Proc. Amer. Math. Soc. 142 (2014), no. 6, 2079-2086.
A. Rybkin.
Spatial Analyticity of solutions to integrable systems. I. The KdV case.
Communications in Partial Differential Equations 38 (2013), no. 5, 802-822.
J. Baggett, O. Bastille, A. Rybkin.
A step-like approximation and a new numerical schema for the Korteweg-de Vries equation in the soliton region.
Electronic Journal of Differential Equations 2013 (2013), no. 40, 1-20.
S. Grudsky, A. Rybkin.
On Toeplitz and Hankel operators with oscillatory symbols containing Blaschke products and applications to the KdV equation.
In: Y. I. Karlovich, L. Rodino, B. Silbermann, and I. M. Spitkovsky (eds.)
Operator Theory, Pseudo-Differential Equations, and Mathematical Physics.
Operator Theory: Advances and Applications, Vol. 228 (2013), Birkhäuser Basel, pp. 127-150.
O. Bastille, A. Rybkin.
On the determinant formula in the inverse scattering procedure with a partially known steplike potential.
Inverse Problems 28 (2012), no. 3, 035001.
A. Rybkin.
The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step-like initial profile decaying on the right half line.
Nonlinearity 24 (2011), no. 10, 2953.
A. Rybkin.
Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line.
Nonlinearity 23 (2010), no. 5, 1143.
A. Rybkin
Regularized Perturbation Determinants and KdV Conservation Laws for Irregular Initial Profiles
In: J. A. Ball, V. Bolotnikov, J. W. Helton, L. Rodman, and I. M. Spitkovsky (eds.).
Topics in Operator Theory, Vol. 2: Systems and mathematical physics.
Operator Theory: Advances and Applications, Vol. 203 (2010), Birkhäuser Basel, pp. 427-444.
A. Rybkin.
On the Marchenko inverse scattering procedure with partial information on the potential.
Inverse Problems 25 (2009), no. 9, 095011.
A. Rybkin, Vu Kim Tuan
A new interpolation formula for the Titchmarsh-Weyl m-function.
Proc. Amer. Math. Soc. 137 (2009), no. 12, 4177-4185.
A. Rybkin.
On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators.
Inverse Problems and Imaging 3 (2009), no. 1, 139-149.
[Preprint]
A. Rybkin.
On the evolution of a reflection coefficient under the KdV flow.
Journal of Mathematical Physics 49 (2008), no. 7, 072701.
[Preprint]
S. Avdonin, V. Mikhaylov, A. Rybkin.
The boundary control approach to the Titchmarsh-Weyl m-function. I. The response operator and the A-amplitude.
Comm. Math. Physics 275 (2007), no. 3, pp. 791-803.
[Preprint]
A. Rybkin.
Preservation of the absolutely continuous spectrum: some extensions of a result by Molchanov-Novitskii-Vainberg.
In: N. Chernov, Y. Karpeshina, I. W. Knowles, R. T. Lewis, and R. Weikard (eds).
Recent Advances in Differential Equations and Mathematical Physics
Contemporary Mathematics Series, Vol. 212 (2006), Amer. Math. Soc., pp. 271-281.
A. Rybkin.
The analytic structure of the reflection coefficient, a sum rule, and a complete description of the Weyl m-function of half-line Schrödinger operators with L2-type potentials.
Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 3, pp. 615-632.
A. Rybkin.
On the spectral L2 conjecture, 3/2-Lieb-Thirring inequality and distributional potentials.
Journal of Mathematical Physics 46 (2005), no. 12, 8 pages.
A. Rybkin.
On a transformation of the Sturm-Liouville equation with slowly decaying potentials and the Titchmarsh-Weyl m-function.
In: J. Janas,P. Kurasov, and S. Naboko (eds.).
Spectral methods for operators of mathematical physics.
Operator Theory: Advances and Applications, Vol. 154 (2004), Birkhäuser Basel, pp. 185-201.
A. Rybkin.
On the absolutely continuous and negative discrete spectra of Schrödinger operators on the line with locally integrable globally square summable potentials.
Journal of Mathematical Physics 45 (2004), no. 4, pp. 1418-1425.
[Preprint]
S. Belov, A. Rybkin.
On the existence of WKB-type asymptotics for the generalized eigenvectors of discrete string operators.
Bull. London Math. Soc. 36 (2004), no. 2, pp. 241-251.
S. M. Belov, A. V. Rybkin.
Higher-order trace formulas of the Buslaev-Faddeev type for the half-line Schrödinger operator with long-range potentials.
Journal of Mathematical Physics 44 (2003), no. 7, pp. 2748-2761.
A. Rybkin.
Necessary and sufficient conditions for absolute summability of the trace formulas for certain one-dimensional Schrödinger operators.
Proceedings of the American Mathematical Society 131 (2003), no. 1, pp. 219-229.
A. Rybkin.
Some new and old asymptotic representation of the Jost solution and Weyl m-function for Schrödinger operators on the line.
Bull. London Math. Soc. 34 (2002), no. 1, pp. 61-72.
A. Rybkin.
A complete analysis of high-energy scattering matrix asymptotics for one dimensional Schrödinger operators with integrable potentials.
Proceedings of the American Mathematical Society 130 (2002), no. 1, pp. 59-67.
A. Rybkin.
On the trace approach to the inverse scattering problem in dimension one.
SIAM Journal of Mathematical Analysis 32 (2001), no. 6, pp. 1248-1264.
[Full text]
A. Rybkin.
KdV-invariants and Herglotz functions.
Differential Integral Equations 14 (2001), no. 4, pp. 493-512.
A. Rybkin.
On an analogue of Cauchy's formula for Hp, ½ ≤ p < 1, and the Cauchy-type integral of a singular measure.
Complex variables and Elliptic Equations 43 (2000), no. 2, pp. 139-149.
A. Rybkin.
On a trace formula of the Buslaev-Faddeev type for a long-range potential.
Journal of Mathematical Physics 40 (1999), no. 3, pp. 1334-1343.
A. V. Rybkin.
On A-integrability of the spectral shift function of unitary operators arising in the Lax-Phillips scattering theory.
Duke Mathematical Journal 83 (1996), no. 3, pp. 683-699.
A. V. Rybkin.
The spectral shift function, the characteristic function of a contraction, and a generalized integral.
Mat. Sbornik 185 (1994), no. 10, pp. 91-144.
English translation in: Russian Acad. of Science Sbornik Math. 83 (1995), no. 1, pp. 237-281.
A. V. Rybkin.
Duality of Lipschitz and Besov classes and the Stieltjes B-integral.
Mat. Zametki 53 (1993), no. 1, pp. 104-110.
English translation in: Mathematical Notes 53 (1993), no. 1, pp. 72-76.
A. V. Rybkin.
On a functional calculus for contractions with absolutely continuous spectrum and an operator Carleson measure.
Ross. Akad. Nauk. Dokl. 326 (1992), no. 3, pp. 417-420.
English translation in: Russian Acad. Sci. Dokl. Math. 46 (1993), no. 2, pp. 282-285.
A. V. Rybkin.
On the evaluation of spectral projectors onto absolutely continuous parts of contraction.
Proceedings of the American Mathematical Society 112 (1991), no. 2, pp. 451-455.
A. V. Rybkin.
Convergence of arguments of Blaschke products in Lp-metrics.
Proceedings of the American Mathematical Society 111 (1991), no. 3, pp. 701-708.
A. V. Rybkin.
The Stieltjes B-integral and the integration of spectral decompositions for some classes of contraction operators.
Dokl. Akad. Nauk SSSR 319 (1991), no. 3, pp. 562-566.
English translation in: Soviet Math. Dokl 44 (1992), no. 1, pp. 166-170.
A. V. Rybkin.
The discrete and the singular spectrum in the trace formula for contractive and unitary operators.
Funktsional. Anal. i Prilozhen. 23 (1989), no. 3, pp. 84-85.
English translation in: Functional Analysis and Applications 23 (1989), no. 3, pp. 242-246.
A. V. Rybkin.
Trace for the pair of a contraction and unitary operators.
Funktsional. Anal. i Prilozh 21 (1987), no. 4, pp. 85-87.
English translation in: Functional Analysis and Applications 21 (1987), no. 4, pp. 334-336.
A. V. Rybkin.
The spectral shift function for a dissipative and a self-adjoint operator, and trace formulas for resonances.
Mat. Sbornik 125 (1984), no. 3, pp. 420-430.
English translation in: Math. USSR Sbornik 53 (1986), no. 2, pp. 421-431.
A. V. Rybkin.
The trace formula for dissipative and selfadjoint operators and spectral identities for resonances.
Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 112 (1984), no. 4, pp. 97-99.
English translation in: Vestnik Leningrad Univ. Math.
A. V. Rybkin.
Trace formulas for resonances.
Teoret. i Matem. Fizhika. 56 (1983), no. 3, pp. 439-447.
English translation in: Theoretical and Math. Physics 56 (1983), no. 3, pp. 927-933.