Geodesic
On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg--de Vries Equation
Matematičeskie zametki, Tome 104 (2018) no. 3, pp. 374-395.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 obtained results are optimal. They are used to study the Cauchy problem for the Korteweg–de Vries equation. Namely, a connection between the smoothness of the solution and the rate of decrease of the initial data at positive infinity is established.
Keywords: Hankel operator, trace-class operator, Korteweg–de Vries equation, inverse problem method. 
@article{MZM_2018_104_3_a4,
     author = {S. M. Grudsky and A. V. Rybkin},
     title = {On the {Trace-Class} {Property} of {Hankel} {Operators} {Arising} in the {Theory} of the {Korteweg--de} {Vries} {Equation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {374--395},
     publisher = {mathdoc},
     volume = {104},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_3_a4/}
}
TY  - JOUR
AU  - S. M. Grudsky
AU  - A. V. Rybkin
TI  - On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg--de Vries Equation
JO  - Matematičeskie zametki
PY  - 2018
SP  - 374
EP  - 395
VL  - 104
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2018_104_3_a4/
LA  - ru
ID  - MZM_2018_104_3_a4
ER  - 
%0 Journal Article
%A S. M. Grudsky
%A A. V. Rybkin
%T On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg--de Vries Equation
%J Matematičeskie zametki
%D 2018
%P 374-395
%V 104
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2018_104_3_a4/
%G ru
%F MZM_2018_104_3_a4
S. M. Grudsky; A. V. Rybkin. On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg--de Vries Equation. Matematičeskie zametki, Tome 104 (2018) no. 3, pp. 374-395. http://geodesic.mathdoc.fr/item/MZM_2018_104_3_a4/

[1] A. Rybkin, “The Hirota $\tau$-function and well-posedness of the KdV equation with an arbitrary step-like initial profile decaying on the right half line”, Nonlinearity, 24:10 (2011), 2953–2990 | DOI | MR | Zbl

[2] V. A. Marchenko, “Nonlinear Equations and Operator Algebras”, Math. Appl. (Soviet Ser.), 17, D. Reidel Publ., Dordrecht, 1988 | MR | Zbl

[3] A. Volberg, P. Yuditskii, “On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length”, Comm. Math. Phys., 226:3 (2002), 567–605 | DOI | MR | Zbl

[4] L. Golinskii, A. Kheifets, F. Peherstorfer, P. Yuditskii, “Scattering theory for CMV matrices: uniqueness, Helson–Szegő and strong Szegő theorems”, Integral Equations Operator Theory, 69:4 (2011), 479–508 | DOI | MR | Zbl

[5] S. Grudsky, A. Rybkin, “Soliton theory and Hankel operators”, SIAM J. Math. Anal., 47:3 (2015), 2283–2323 | DOI | MR | Zbl

[6] A. Rybkin, “KdV equation beyond standard assumptions on initial data”, Phys. D, 365 (2018), 1–11 | DOI | MR | Zbl

[7] D. V. Zakharov, S. A. Dyachenko, V. E. Zakharov, “Bounded solutions of KdV and non-periodic one-gap potentials in quantum mechanics”, Lett. Math. Phys., 106:6 (2016), 731–740 | DOI | MR | Zbl

[8] D. V. Zakharov, S. A. Dyachenko, V. E. Zakharov, “Primitive potentials and bounded solutions of the KdV equation”, Phys. D, 333 (2016), 148–156 | DOI | MR

[9] A. Rybkin, “Spatial analyticity of solutions to integrable systems. I. The KdV case”, Comm. Partial Differential Equations, 38:5 (2013), 802–822 | DOI | MR | Zbl

[10] A. Cohen, T. Kappeler, “Solutions to the Korteweg–de Vries equation with initial profile in $L^1_1(R)\cap L_{N^1}(R^+)$”, SIAM J. Math. Anal., 18:4 (1987), 991–1025 | DOI | MR | Zbl

[11] V. V. Peller, “Operatory Gankelya klassa $\mathfrak S_p$ i ikh prilozheniya (ratsionalnaya approksimatsiya, gaussovskie protsessy, problema mazhoratsii operatorov)”, Matem. sb., 113 (155):4 (12) (1980), 538–581 | MR | Zbl

[12] V. V. Peller, Hankel Operators and Their Applications, Springer, New York, 2003 | MR | Zbl

[13] M. V. Fedoryuk, Metod perevala, Nauka, M., 1977 | MR | Zbl

[14] S. P. Novikov, S. V. Manakov, L. P. Pitaevski{ĭ}, V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Plenum, New York, 1984 | MR | Zbl

[15] P. Deift, E. Trubowitz, “Inverse scattering on the line”, Comm. Pure Appl. Math., 32:2 (1979), 121–251 | DOI | MR | Zbl

[16] B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Nelineinye uravneniya tipa Kortevega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, UMN, 31:1 (187) (1976), 55–136 | MR | Zbl

[17] F. Gesztesy, H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Vol. I. $(1+1)$-Dimensional Continuous Models, Cambridge Stud. Adv. Math., 79, Cambridge Univ. Press, Cambridge, 2003 | MR

[18] A. R. Its, V. B. Matveev, “Operatory Shredingera s konechnozonnym spektrom i $N$-solitonnye resheniya uravneniya Kortevega–de Frisa”, TMF, 23:1 (1975), 51–68 | MR

[19] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, “Method for solving the Korteweg–de Vries equation”, Phys. Rev. Lett., 19 (1967), 1095–1097 | DOI | Zbl

[20] I. Krichever, S. P. Novikov, “Periodic and almost-periodic potentials in inverse problems”, Inverse Problems, 15:6 (1999), R117–R144 | DOI | MR | Zbl

[21] J. B. McLeod, P. J. Olver, “The connection between partial differential equations soluble by inverse scattering and ordinary differential equations of Painlevé type”, SIAM J. Math. Anal., 14:3 (1983), 488–506 | DOI | MR | Zbl

[22] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser., 149, Cambridge Univ. Press, Cambridge, 1991 | MR | Zbl

[23] V. A. Marchenko, “The Cauchy problem for the KdV equation with nondecreasing initial data”, What is integrability?, Springer Ser. Nonlinear Dynam., Springer-Verlag, Berlin, 1991, 273–318 | DOI | MR | Zbl

[24] Percy Deift, “Some open problems in random matrix theory and the theory of integrable systems”, Integrable Systems and Random Matrices, Contemp. Math., 458, Amer. Math. Soc., Providence, RI, 2008, 419–430 | DOI | MR | Zbl

[25] P. Dubard, P. Gaillard, C. Klein, V. B. Matveev, “On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation”, Eur. Phys. J. Special Topics, 185:1 (2010), 247–258 | DOI

[26] A. V. Gurevich, P. Pitaevskii, “Raspad nachalnogo razryva v uravnenii Kortevega–de Friza”, Pisma v ZhETF, 17:5 (1973), 193–195

[27] E. Ya. Khruslov, “Asimptotika resheniya zadachi Koshi dlya uravneniya Kortevega–de Friza s nachalnymi dannymi tipa stupenki”, Matem. sb., 99 (141):2 (1976), 261–281 | MR | Zbl

[28] E. Ya. Khruslov, V. P. Kotlyarov, “Soliton asymptotics of nondecreasing solutions of nonlinear completely integrable evolution equations”, Spectral Operator Theory and Related Topics, Adv. Soviet Math., 19, Amer. Math. Soc., Providence, RI, 1994, 129–180 | MR | Zbl

[29] A. Rybkin, “On Peller's characterization of trace class Hankel operators and smoothness of KdV solutions”, Proc. Amer. Math. Soc., 146:4 (2018), 1627–1637 | MR | Zbl

[30] R. Hirota, “Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27 (1971), 1192–1194 | DOI | Zbl