×
Grudsky, Sergei; Rybkin, Alexei

On positive type initial profiles for the KdV equation. (English) Zbl 1294.35122

Proc. Am. Math. Soc. 142, No. 6, 2079-2086 (2014).
The paper addresses the problem of well-posedness for the classical Korteweg- de Vries (KdV) equation with the locally integrable initial condition of the step-like shape, given by the analytical form \(q(x) = dr/dx + r^2\), which is related to the form of the Miura transformation. This step-like profile does not vanish in one direction at the spatial infinity. Using the technique of the Hankel operators, it is proved that any initial condition of this type evolves into a meromorphic functions without real poles. In particular, it cannot develop the so-called positons, i.e., singular solitons.
Reviewer: Boris A. Malomed (Tel Aviv)

Cited in 9 Documents

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)

Keywords:

well-posedness; positon; Miura transform; step-like initial conditions; meromorphic functions; Hankel operator; Titchmarsh-Weyl functions
Cite Review PDF
Full Text: DOI arXiv

References:

[1] A. Böttcher, S. Grudsky, and I. Spitkovsky, Toeplitz operators with frequency modulated semi-almost periodic symbols, J. Fourier Anal. Appl. 7 (2001), no. 5, 523 – 535. · Zbl 0996.47035 · doi:10.1007/BF02511224
[2] Albrecht Böttcher and Bernd Silbermann, Analysis of Toeplitz operators, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Prepared jointly with Alexei Karlovich. · Zbl 1098.47002
[3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Sharp global well-posedness for KdV and modified KdV on \Bbb R and \?, J. Amer. Math. Soc. 16 (2003), no. 3, 705 – 749. · Zbl 1025.35025
[4] Vladimir Dybin and Sergei M. Grudsky, Introduction to the theory of Toeplitz operators with infinite index, Operator Theory: Advances and Applications, vol. 137, Birkhäuser Verlag, Basel, 2002. Translated from the Russian by Andrei Iacob. · Zbl 1030.47001
[5] Iryna Egorova, Katrin Grunert, and Gerald Teschl, On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data. I. Schwartz-type perturbations, Nonlinearity 22 (2009), no. 6, 1431 – 1457. · Zbl 1171.35103 · doi:10.1088/0951-7715/22/6/009
[6] Sergei M. Grudsky, Toeplitz operators and the modelling of oscillating discontinuities with the help of Blaschke products, Problems and methods in mathematical physics (Chemnitz, 1999) Oper. Theory Adv. Appl., vol. 121, Birkhäuser, Basel, 2001, pp. 162 – 193. · Zbl 1047.47021
[7] Thomas Kappeler, Peter Perry, Mikhail Shubin, and Peter Topalov, The Miura map on the line, Int. Math. Res. Not. 50 (2005), 3091 – 3133. · Zbl 1089.35058 · doi:10.1155/IMRN.2005.3091
[8] Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. · Zbl 1007.47001
[9] V. B. Matveev, Positons: slowly decreasing analogues of solitons, Teoret. Mat. Fiz. 131 (2002), no. 1, 44 – 61 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 131 (2002), no. 1, 483 – 497. · Zbl 1029.37051 · doi:10.1023/A:1015149618529
[10] Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. · Zbl 1030.47002
[11] Alexei Rybkin, Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line, Nonlinearity 23 (2010), no. 5, 1143 – 1167. · Zbl 1188.37068 · doi:10.1088/0951-7715/23/5/007
[12] Alexei Rybkin, The Hirota \?-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 – 2990. · Zbl 1229.49026 · doi:10.1088/0951-7715/24/10/015
[13] Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. · Zbl 1106.35001
[14] Gerald Teschl, Mathematical methods in quantum mechanics, Graduate Studies in Mathematics, vol. 99, American Mathematical Society, Providence, RI, 2009. With applications to Schrödinger operators. · Zbl 1166.81004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.