Abstract
In this chapter, partially following Leble (Optical Solitons. Theoretical and Experimental Challenges. Springer, Berlin, pp. 71–104, 2003) [1], we sketch the basic mathematical tools used in the theory of integrable systems embedded into the waveguide propagation of nonlinear waves, starting from general relations and illustrating them by simple examples.
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References
S. Leble, Nonlinear waves in optical waveguides and Soliton theory applications, in Optical Solitons. Theoretical and Experimental Challenges. (Springer, Berlin, 2003), pp. 71–104
S.P. Novikov, S.V. Manakov, L.P. Pitaevski, V.E. Zakharov, Theory of Solitons (Plenum, New York, 1984)
S. Leble, Nonlinear Waves in Waveguides (Springer, Berlin, 1991)
V.E. Zakharov, S. Wabnitz (eds.), Optical Solitons, Theoretical Challenges and Industrial Perspectives, Les Houches Workshop, 1998 (Springer, Berlin, 1999)
V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991)
S. Leble, Elementary and binary Darboux transformations at rings. Comput. Math. Appl. 35, 73 (1998)
S. Leble, Theor. Math. Phys. 22, 239 (2000)
D.J. Kaup, The first-order perturbed SBS equations. J. Nonlinear Sci. 3, 427 (1993)
I. Leonhardt, H. Steudel, Evolution of solitons in stimulated raman-scattering from cos-shaped initial pulses. Appl. Phys. B 60, S221–S225 (1995)
J. Leon, A. Mikhailov, Raman soliton generation from laser inputs in SRS. Phys. Lett. A 253, 33 (1999)
V. Cautaerts, Y. Kodama, A. Maruta, H. Sugavara: Nonlinear Pulses in Ultra-Fast Communications. Les Houches Lectures, Lecture 9. (Springer, Berlin, 1999), p. 147
E. Doktorov, S.B. Leble, Dressing Method in Mathematical Physics (Springer, Berlin, 2007)
S.B. Leble, N.V. Ustinov, Solitons of nonlinear equations associated with degenerate spectral problem of the third order, in Nonlinear Theory and Its Applications (NOLTA’93), ed. by M. Tanaka, T. Saito (World Scientific, Singapore, 1993), pp. 547–550
A.V. Mikhailov, The reduction problem and the inverse scattering method. Physica D 3, 73 (1981)
V. Gerdjikov, A. Yanovski, Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system. J. Math. Phys. 35, 3687 (1994)
S.B. Leble, N.V. Ustinov, On soliton and periodic solutions of Maxwell-Bloch system for two-level medium with degeneracy. CSF 11, 1763 (2000). arXiv:quant-ph/9810049
A.M. Basharov, A.I. Maimistov, JETP 87, 1595 (1984)
J. Eerkens, J.F. Kunze, L.J. Bond, Laser isotope encrichment for medical and industrial applications. (January 2006). https://doi.org/10.1115/ICONE14-89767
L.A. Bolshov, N.N. Elkin, V.V. Likhansky, M.I. Persiantsev, The theory of the coherent transformation of the frequency of ultrashort light pulses in resonant media. JETP 94, 101 (1988)
S. Leble, N.V. Ustinov, The third order spectral problem reductions and darboux transformations. Inverse Probl. 10, 617 (1994)
M. Kuna, M. Czachor, S. Leble, Nonlinear von Neumann-type equations: darboux invariance and spectra. Phys. Lett. A 255, 42 (1999)
H. Steudel, in Proceedings of 3rd International Workshop on Nonlinear Processes in Physics, ed. by V.G. Bar’yakhtar et al., vol. 1. (Kiev, Naukova Dumka, 1988), p. 144
H. Steudel, N-soliton solutions to degenerate self-induced transparency. J. Mod. Opt. 35, 693 (1988)
S. Leble, Integrable models for density matrix evolution, in: Proceedings of the workshop on Nonlinearity, Integrability and all that: Twenty Years after NEEDS’79, ed. by M. Boiti et al. (World Scientific, 2000) pp. 311–317
S.B. Leble, M. Czachor, Darboux-integrable nonlinear Liouville-von Neumann equation. Phys. Rev. E 58, 7091–7100 (1998). arXiv:quant-ph/9804052
S. Manakov: Funktsional’nyi Analiz i Pril. 10, 93 (1976)
M. Adler, P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math. 38, 267 (1980)
N. Ustinov, S. Leble, M. Czachor, M. Kuna, Darboux-integration of \(\imath \rho _t = [H, f(\rho )]\). Phys. Lett. A 279, 333 (2001). arXiv:quant-ph/0005030
M. Czachor, S. Leble, M. Kuna, J. Naudts, Nonlinear von Neumann type equations, in: Proceedings of the International Symposium on Trends in Quantum Mechanics, ed. H.-D. Doebner et al. (World Scientific, 2000) pp. 209–226
H. Steudel, D.J. Kaup, Degenerate two-photon propagation and the oscillating two-stream instability: the general solution for amplitude-modulated pulses. J. Mod. Opt 43, 1851–1866 (1996)
R.K. Boullough, P.M. Jack, P.W. Kitchenside, R. Saunders, Solitons in laser physics. Physica Scripta 20, 364 (1979)
M. Chbat, C. Menyuk, I. Glesk, P. Prucnal, Interactions of bound multiple solitons in strongly birefringent fibers. Opt. Lett. 20, 258 (1995)
R.A. Vlasov, E.V. Doktorov, Covariant methods in theoretical physics, optics and acoustics. 94 Minsk (1981)
S. Kakei, J. Satsuma, Multi-soliton solutions of a coupled system of the nonlinear Schrodinger equation and the Maxwell-Bloch equations. J. Phys. Soc. Jpn. 63, 885 (1994)
R.K. Boullough, F. Ahmad, Exact solutions of the self-induced transparency equations. Phys. Rev. Lett. 27, 330 (1971)
J.U. Kang, G.I. Stegeman, J.S. Atchison, One-dimensional spatial soliton dragging, trapping, and all-optical switching in AlGaAs waveguides. Opt. Lett. 21, 189 (1996)
P.L. Christiansen, J.C. Eilbeck, V.Z. Enolski, N.A. Kostov, Quasi-periodic and periodic solutions for coupled nonlinear Schrodinger equations of Manakov type. Proc. R. Soc. Lond. A 456, 2263 (2000)
H. Steudel, Annalen der Physik (Leipzig) 32(205), 445 (1975)
D.J. Kaup, Perturbation theory for solitons in optical fibers. Phys. Rev. A 42, 5689 (1990)
J. Yang, Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics. Phys. Rev. E 59, 2393 (1999)
V. Shchesnovich, E. Doctorov, Modified Manakov system with self-consistent source. Phys. Lett. A 213, 23 (1996)
C. Anastassiu, M. Segev, K. Steiglitz et al., Energy-exchange interactions between colliding vector solitons. PRL 83, 2332 (1999)
V. Gerdjikov et al., Stability and quasi-equidistant propagation of NLS soliton trains. Phys. Lett. A 241, 323 (1998)
A.I. Maimistov, E.A. Manykin, Propagation of ultrashort optical pulses in resonant non-linear light guides. Sov. Phys. JETP 58, 685 (1983)
S. Chakravarty, Soliton solutions of coupled Maxwell–Bloch equations. Phys. Lett. A 380(11–12), 1141–1150 (2016)
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Leble, S. (2019). Solitonics. In: Waveguide Propagation of Nonlinear Waves. Springer Series on Atomic, Optical, and Plasma Physics, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-030-22652-7_5
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DOI: https://doi.org/10.1007/978-3-030-22652-7_5
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