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The sine-Gordon equation in the semiclassical limit: Dynamics of fluxon condensates
About this Title
Robert J. Buckingham, Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, Ohio 45221 and Peter D. Miller, Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, Michigan 48109
Publication: Memoirs of the American Mathematical Society
Publication Year:
2013; Volume 225, Number 1059
ISBNs: 978-0-8218-8545-1 (print); 978-1-4704-1059-9 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00672-1
Published electronically: February 20, 2013
Keywords: Sine-Gordon,
semiclassical asymptotics,
Riemann-Hilbert problems
MSC: Primary 35Q51, 35B40, 35Q15
Table of Contents
Chapters
- 1. Introduction
- 2. Formulation of the Inverse Problem for Fluxon Condensates
- 3. Elementary Transformations of J$(w)$
- 4. Construction of $g(w)$
- 5. Use of $g(w)$
- A. Proofs of Propositions Concerning Initial Data
- B. Details of the Outer Parametrix in Cases $\mathsf {L}$ and $\mathsf {R}$
Abstract
We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. We show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham’s formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases.- N. I. Akhiezer, Elements of the theory of elliptic functions, Translations of Mathematical Monographs, vol. 79, American Mathematical Society, Providence, RI, 1990. Translated from the second Russian edition by H. H. McFaden. MR 1054205
- J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin, and P. D. Miller, Discrete orthogonal polynomials, Annals of Mathematics Studies, vol. 164, Princeton University Press, Princeton, NJ, 2007. Asymptotics and applications. MR 2283089
- A. Barone, F. Esposito, C. J. Magee, and A. C. Scott, “Theory and applications of the sine-Gordon equation,” Riv. Nuovo Cim., 1, 227–267, 1971.
- Robert Buckingham and Peter D. Miller, Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation, Phys. D 237 (2008), no. 18, 2296–2341. MR 2455608, DOI 10.1016/j.physd.2008.02.010
- R. Buckingham and P. D. Miller, “The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix”, http://arxiv.org/abs/1106.5716, 2011. To appear in J. Anal. Math.
- P. Deift, S. Venakides, and X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Internat. Math. Res. Notices 6 (1997), 286–299. MR 1440305, DOI 10.1155/S1073792897000214
- P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295–368. MR 1207209, DOI 10.2307/2946540
- P. A. Deift and X. Zhou, Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math. 48 (1995), no. 3, 277–337. MR 1322812, DOI 10.1002/cpa.3160480304
- B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11–80 (Russian). With an appendix by I. M. Krichever. MR 616797
- Nicholas Ercolani, M. Gregory Forest, and David W. McLaughlin, Modulational stability of two-phase sine-Gordon wavetrains, Stud. Appl. Math. 71 (1984), no. 2, 91–101. MR 760226, DOI 10.1002/sapm198471291
- N. Ercolani, M. G. Forest, D. W. McLaughlin, and R. Montgomery, Hamiltonian structure for the modulation equations of a sine-Gordon wavetrain, Duke Math. J. 55 (1987), no. 4, 949–983. MR 916131, DOI 10.1215/S0012-7094-87-05548-7
- M. Gregory Forest and David W. McLaughlin, Modulations of sinh-Gordon and sine-Gordon wavetrains, Stud. Appl. Math. 68 (1983), no. 1, 11–59. MR 686249, DOI 10.1002/sapm198368111
- Herbert Goldstein, Classical mechanics, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1980. Addison-Wesley Series in Physics. MR 575343
- Spyridon Kamvissis, Kenneth D. T.-R. McLaughlin, and Peter D. Miller, Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation, Annals of Mathematics Studies, vol. 154, Princeton University Press, Princeton, NJ, 2003. MR 1999840
- M. Klaus and J. K. Shaw, Purely imaginary eigenvalues of Zakharov-Shabat systems, Phys. Rev. E (3) 65 (2002), no. 3, 036607, 5. MR 1905252, DOI 10.1103/PhysRevE.65.036607
- Igor Moiseevich Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Uspehi Mat. Nauk 32 (1977), no. 6(198), 183–208, 287 (Russian). MR 0516323
- Gregory D. Lyng and Peter D. Miller, The $N$-soliton of the focusing nonlinear Schrödinger equation for $N$ large, Comm. Pure Appl. Math. 60 (2007), no. 7, 951–1026. MR 2319053, DOI 10.1002/cpa.20162
- Peter D. Miller, Asymptotics of semiclassical soliton ensembles: rigorous justification of the WKB approximation, Int. Math. Res. Not. 8 (2002), 383–454. MR 1884077, DOI 10.1155/S1073792802109020
- Peter D. Miller and Spyridon Kamvissis, On the semiclassical limit of the focusing nonlinear Schrödinger equation, Phys. Lett. A 247 (1998), no. 1-2, 75–86. MR 1650432, DOI 10.1016/S0375-9601(98)00565-9
- Junkichi Satsuma and Nobuo Yajima, Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Progr. Theoret. Phys. Suppl. No. 55 , posted on (1974), 284–306. MR 0463733, DOI 10.1143/ptps.55.284
- A. C. Scott, “Waveform stability on a nonlinear Klein-Gordon equation,” Proc. IEEE (Lett.) 57, 1338–1339, 1969.
- A. C. Scott, F. Chu, S. Reible, “Magnetic flux propagation on a Josephson transmission line,” J. Appl. Phys. 47, 3272–3286, 1976.
- Alexander Tovbis, Stephanos Venakides, and Xin Zhou, On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation, Comm. Pure Appl. Math. 57 (2004), no. 7, 877–985. MR 2044068, DOI 10.1002/cpa.20024
- G. B. Whitham, Non-linear dispersive waves, Proc. Roy. Soc. London Ser. A 283 (1965), 238–261. MR 176724, DOI 10.1098/rspa.1965.0019
- G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
- http://functions.wolfram.com/EllipticFunctions/EllipticTheta1/18/02/02/01/.