×

zbMATH — the first resource for mathematics

Real-analytic solutions of the nonlinear Schrödinger equation. (English. Russian original) Zbl 1318.35105
Trans. Mosc. Math. Soc. 2014, 173-183 (2014); translation from Tr. Mosk. Mat. O.-va 75, No. 2, 205-218 (2014).
Summary: We establish that the Riemann problem on the factorization of formal matrix-valued Laurent series subject to unitary symmetry has a solution. As an application, we show that any local real-analytic solution (in \( x\) and \( t\)) of the focusing nonlinear Schrödinger equation has a real-analytic extension to some strip parallel to the \( x\)-axis and that in each such strip there exists a solution that cannot be extended further.
MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118 – 134 (Russian, with English summary); English transl., Soviet Physics JETP 34 (1972), no. 1, 62 – 69.
[2] Гамил\(^{\приме}\)тонов подход в теории солитонов, ”Наука”, Мосцощ, 1986 (Руссиан). · Zbl 0632.58003
[3] N. N. Akhmediev and A. Ankevich, Solitons: nonlinear pulses and beams, Chapman and Hall, London, 1997; Russian transl., Fizmatlit, Moscow, 2003.
[4] A. R. Osborne, Classification of homoclinic rogue wave solutions of the nonlinear Schrödinger equation, Nat.Hazards Earth Syst. Sci. Discuss. 2 (2014), 897-933.
[5] A. O. Smirnov and G. M. Golovachev, Three-phase solutions of the nonlinear Schrödinger equation in terms of elliptic functions, Nelineinaya dinamika 9 (2013), no. 3, 389-407. (Russian)
[6] A. V. Domrin, The Riemann problem and matrix-valued potentials with a converging Baker-Akhiezer function, Teoret. Mat. Fiz. 144 (2005), no. 3, 453 – 471 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 144 (2005), no. 3, 1264 – 1278. · Zbl 1178.30048 · doi:10.1007/s11232-005-0158-y · doi.org
[7] A. V. Domrin, Meromorphic extension of solutions of soliton equations, Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), no. 3, 23 – 44 (Russian, with Russian summary); English transl., Izv. Math. 74 (2010), no. 3, 461 – 480. · Zbl 1202.35186 · doi:10.1070/IM2010v074n03ABEH002494 · doi.org
[8] I. V. Cherednik, Regularity of ”finite-zone” solutions of integrable matrix differential equations, Dokl. Akad. Nauk SSSR 266 (1982), no. 3, 593 – 597 (Russian). · Zbl 0552.35017
[9] I. C. Gohberg and M. G. Kreĭn, Systems of integral equations on the half-line with kernels depending on the difference of the arguments, Uspehi Mat. Nauk (N.S.) 13 (1958), no. 2 (80), 3 – 72 (Russian).
[10] A. V. Domrin, Remarks on a local version of the method of the inverse scattering problem, Tr. Mat. Inst. Steklova 253 (2006), no. Kompleks. Anal. i Prilozh., 46 – 60 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 2(253) (2006), 37 – 50. · Zbl 1351.35096
[11] Спектрал\(^{\приме}\)ные преобразования и солитоны, ”Мир”, Мосцощ, 1985 (Руссиан). Методы решения и исследования нелинейных ѐволюционных уравнений. [Тоолс то солве анд инвестигате нонлинеар еволутион ечуатионс]; Транслатед фром тхе Енглиш бы М. А. Ол\(^{\приме}\)шанецкий анд Н. Т. Пащенко; Транслатион едитед анд щитх а префаце бы В. Е. Захаров.
[12] A. R. It\cdot s, A. V. Rybin, and M. A. Sall\(^{\prime}\), On the exact integration of the nonlinear Schrödinger equation, Teoret. Mat. Fiz. 74 (1988), no. 1, 29 – 45 (Russian, with English summary); English transl., Theoret. and Math. Phys. 74 (1988), no. 1, 20 – 32. · Zbl 0678.35012 · doi:10.1007/BF01018207 · doi.org
[13] Andrew N. W. Hone, Crum transformation and rational solutions of the non-focusing nonlinear Schrödinger equation, J. Phys. A 30 (1997), no. 21, 7473 – 7483. · Zbl 0928.35164 · doi:10.1088/0305-4470/30/21/019 · doi.org
[14] A. V. Domrin, On holomorphic solutions of equations of Korteweg – de Vries type, Trans. Moscow Math. Soc. , posted on (2012), 193 – 206. · Zbl 1291.35278
[15] A. V. Komlov, On the poles of Picard potentials, Tr. Mosk. Mat. Obs. 71 (2010), 270 – 282 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2010), 241 – 250. · Zbl 1218.34102 · doi:10.1090/S0077-1554-2010-00182-3 · doi.org
[16] Функционал\(^{\приме}\)ный анализ., Издат. ”Мир”, Мосцощ, 1975 (Руссиан). Транслатед фром тхе Енглиш бы В. Ја. Лин; Едитед бы Е. А. Горин.
[17] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, (Vol. II by R. Courant.), Interscience Publishers (a division of John Wiley & Sons), New York-Lon don, 1962. · Zbl 0099.29504
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.