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Solution of paraxial wave equation for inhomogeneous media in linear and quadratic approximation


Authors: Alex Mahalov and Sergei K. Suslov
Journal: Proc. Amer. Math. Soc. 143 (2015), 595-610
MSC (2010): Primary 35Q55, 35Q51; Secondary 35C05, 81Q05
DOI: https://doi.org/10.1090/S0002-9939-2014-12295-7
Published electronically: October 28, 2014
MathSciNet review: 3283647
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Abstract: We construct explicit solutions of the inhomogeneous parabolic wave equation in a linear and quadratic approximation. As examples, oscillating laser beams in a $ 1D$ parabolic waveguide, spiral light beams in $ 2D$ varying media and an effect of superfocusing of particle beams in a thin monocrystal film are briefly discussed. Transformations of nonlinear equations into the corresponding autonomous and homogeneous forms are found and a review of important applications is also given.


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  • [1] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, vol. 149, Cambridge University Press, Cambridge, 1991. MR 1149378 (93g:35108)
  • [2] Mark J. Ablowitz and Harvey Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 642018 (84a:35251)
  • [3] E. G. Abramochkin and V. G. Volostnikov, Spiral light beams, Physics-Uspekhi 47 (2004), no. 12, 1177-1203.
  • [4] G. P. Agrawal, A. K. Ghatak, and C. L. Mehtav, Propagation of a partially coherent beam through selfoc fibers, Opt. Comm. 12 (1974), no. 3, 333-337.
  • [5] S. A. Akhmanov, Yu. E. Dyakov, and A. S. Chirkin, Vvedenie v statisticheskuyu radiofiziku i optiku, Nauka, Moscow, 1981 (Russian). MR 626992 (83c:78001)
  • [6] S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium, Physics-Uspekhi 10 (1968), no. 1-2, 609-636.
  • [7] S. A. Akhmanov, V. A. Vyslouh, and A. S. Chirkin, Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation, Physics-Uspekhi 29 (1986), no. 7, 642-677.
  • [8] S. A. Akhmanov, V. A. Vyslouh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses, Nauka, Moscow, 1988 [in Russian].
  • [9] M. V. Berry and N. L. Balazs, Nonspreading wave packets, Am. J. Phys. 47 (1979), no. 2, 264-267.
  • [10] H. H. Chen, Y. C. Lee, and C. S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method, Special issue on solitons in physics. Phys. Scripta 20 (1979), no. 3-4, 490-492. MR 544493 (81e:35008), https://doi.org/10.1088/0031-8949/20/3-4/026
  • [11] Hsing Hen Chen and Chuan Sheng Liu, Solitons in nonuniform media, Phys. Rev. Lett. 37 (1976), no. 11, 693-697. MR 0411488 (53 #15222)
  • [12] H.-H. Chen and Ch.-Sh. Liu, Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities, Phys. Fluids 21 (1978), no. 3, 377-380.
  • [13] S. Roy Choudhury, Modulated amplitude waves in the cubic-quintic Ginzburg-Landau equation, Math. Comput. Simulation 69 (2005), no. 3-4, 243-256. MR 2152257, https://doi.org/10.1016/j.matcom.2005.01.003
  • [14] P. A. Clarkson, Painlevé analysis of the damped, driven nonlinear Schrödinger equation, Proc. Roy. Soc. Edin., 109A (1988), 109-126. MR 0952332 (89k:35200)
  • [15] Peter A. Clarkson, Dimensional reductions and exact solutions of a generalized nonlinear Schrödinger equation, Nonlinearity 5 (1992), no. 2, 453-472. MR 1158381 (93f:35212)
  • [16] NIST handbook of mathematical functions, U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010. Edited by Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark; with 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248 (2012a:33001)
  • [17] Peter A. Clarkson and Christopher M. Cosgrove, Painlevé analysis of the nonlinear Schrödinger family of equations, J. Phys. A 20 (1987), no. 8, 2003-2024. MR 893304 (89c:35136)
  • [18] Robert Conte and Micheline Musette, Linearity inside nonlinearity: exact solutions to the complex Ginzburg-Landau equation, Phys. D 69 (1993), no. 1-2, 1-17. MR 1245653 (94m:35281), https://doi.org/10.1016/0167-2789(93)90177-3
  • [19] R. Conte and M. Musette, Elliptic general analytic solutions, Stud. Appl. Math. 123 (2009), no. 1, 63-81. MR 2538286 (2010m:35053), https://doi.org/10.1111/j.1467-9590.2009.00447.x
  • [20] Robert Conte and Tuen-Wai Ng, Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation, Acta Appl. Math. 122 (2012), 153-166. MR 2993990, https://doi.org/10.1007/s10440-012-9734-y
  • [21] R. Conte and T.-W. Ng, Detection and construction of an elliptic solution of the complex cubic-quintic Ginzburg-Landau equation, Theor. Math. Phys. 172 (2012), no. 2, 1073-1084.
  • [22] Ricardo Cordero-Soto, Raquel M. Lopez, Erwin Suazo, and Sergei K. Suslov, Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields, Lett. Math. Phys. 84 (2008), no. 2-3, 159-178. MR 2415547 (2009m:81055), https://doi.org/10.1007/s11005-008-0239-6
  • [23] R. Kordero-Soto and S. K. Suslov, Time reversal for modified oscillators, Teoret. Mat. Fiz. 162 (2010), no. 3, 345-380 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 162 (2010), no. 3, 286-316. MR 2682129 (2011h:81100), https://doi.org/10.1007/s11232-010-0023-5
  • [24] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71 (1999), 463-512.
  • [25] Yu. N. Demkov, Channeling, superfocusing, and nuclear reactions, Phys. Atom. Nuclei, 72 (2009), no. 5, 779-785.
  • [26] Yu. N. Demkov and J. D. Meyer, A sub-atomic microscope, superfocusing in channeling and close encounter atomic and nuclear reactions, Eur. Phys. J. B 42 (2004), 361-365.
  • [27] V. V. Dodonov and V. I. Manko, Invariants and correlated states of nonstationary quantum systems, Trudy Fiz. Inst. Lebedev. 183 (1987), 71-181, 289 (Russian). MR 937883 (89i:81044)
  • [28] Franco Flandoli and Alex Mahalov, Stochastic three-dimensional rotating Navier-Stokes equations: averaging, convergence and regularity, Arch. Ration. Mech. Anal. 205 (2012), no. 1, 195-237. MR 2927621, https://doi.org/10.1007/s00205-012-0507-6
  • [29] V. A. Fock, Selected works. Quantum mechanics and quantum field theory, edited by L. D. Faddeev, L. A. Khalfin and I. V. Komarov. Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2083062 (2005m:81003)
  • [30] V. A. Fock, Electromagnetic diffraction and propagation problems, International Series of Monographs on Electromagnetic Waves, Vol. 1, Pergamon Press, Oxford, 1965. MR 0205569 (34 #5396)
  • [31] L. Gagnon, Exact traveling-wave solutions for optical models based on the nonlinear cubic-quintic Schrödinger equation, J. Opt. Soc. Amer. A 6 (1989), no. 9, 1477-1483. MR 1015382 (90g:78006), https://doi.org/10.1364/JOSAA.6.001477
  • [32] L. Gagnon, G. Grammaticos, A. Ramani, and P. Winternitz, Lie symmetries of a generalised non-linear Schrödinger equation: III. Reductions to third-order ordinary differential equations, J. Phys. A: Math. Gen. 22 (1989), 499-509. MR 0984527 (90b:35014)
  • [33] L. Gagnon and P. Winternitz, Lie symmetries of a generalised nonlinear Schrödinger equation. I. The symmetry group and its subgroups, J. Phys. A 21 (1988), no. 7, 1493-1511. MR 951040 (89g:58220)
  • [34] L. Gagnon and P. Winternitz, Lie symmetries of a generalised nonlinear Schrödinger equation. II. Exact solutions, J. Phys. A 22 (1989), no. 5, 469-497. MR 984526 (90b:35013)
  • [35] L. Gagnon and P. Winternitz, Symmetry classes of variable coefficient nonlinear Schrödinger equations, J. Phys. A 26 (1993), no. 23, 7061-7076. MR 1253895 (95f:35238)
  • [36] J. A. Giannini and R. I. Joseph, The role of the second Painlevé transcendent in nonlinear optics, Phys. Lett. A 141 (1989), no. 8-9, 417-419. MR 1030927 (90i:78038), https://doi.org/10.1016/0375-9601(89)90860-8
  • [37] A. V. Gurevich, Nonlinear phenomena in the ionosphere, Springer-Verlag, Berlin, 1978.
  • [38] A. N. W. Hone, Painlevé tests, singularity structure and integrability, Lecture Notes in Phys., 767, Springer, Berlin, 2009. MR 2867552 (2012k:34199)
  • [39] A. N. W. Hone, Non-existence of elliptic travelling wave solutions of the complex Ginzburg-Landau equation, Phys. D 205 (2005), no. 1-4, 292-306. MR 2167157 (2006e:35302), https://doi.org/10.1016/j.physd.2004.10.011
  • [40] R. S. Johnson, On the modulation of water waves in the neighbourhood of $ kh\approx 1.363$, Proc. Roy. Soc. London Ser. A 357 (1977), no. 1689, 131-141. MR 0479013 (57 #18469)
  • [41] B. B. Kadomtsev and V. I. Karpman, Nonlinear waves, Uspehi Fiz. Nauk 103 (1971), 193-232 (Russian); English transl., Soviet Physics Uspekhi 14 (1971), 40-60. MR 0449148 (56 #7453)
  • [42] Yu. Kagan, E. L. Surkov and G. V. Shlyapnikov, Evolution and global collapse of trapped Bose condensates under variations of the scattering length, Phys. Rev. Lett. 79 (1997), no. 14, 2604-2607.
  • [43] Saburo Kakei, Narimasa Sasa, and Junkichi Satsuma, Bilinearization of a generalized derivative nonlinear Schrödinger equation, J. Phys. Soc. Japan 64 (1995), no. 5, 1519-1523. MR 1335407 (96a:35195), https://doi.org/10.1143/JPSJ.64.1519
  • [44] Christian Kharif, Efim Pelinovsky, and Alexey Slunyaev, Rogue waves in the ocean, Advances in Geophysical and Environmental Mechanics and Mathematics, Springer-Verlag, Berlin, 2009. MR 2841079 (2012g:86006)
  • [45] E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, Low-dimensional Bose liquids: beyond the Gross-Pitaevskii approximation, Phys. Rev. Lett. 85 (2000), no. 6, 1146-1149.
  • [46] C. Krattenthaler, S. I. Kryuchkov, A. Mahalov, and S. K. Suslov, On the problem of electromagnetic-field quantization, arXiv:1301.7328v2 [math-ph] 9 Apr 2013.
  • [47] A. Kundu, Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations, J. Math. Phys. 25 (1984), no. 12, 3433-3438. MR 767547 (86c:58069), https://doi.org/10.1063/1.526113
  • [48] E. A. Kuznetsov and S. K. Turitsyn, Talanov transformations in self-focusing problems and instability of stationary waveguides, Phys. Lett. A 112 (1985), no. 6-7, 273-275.
  • [49] N. Lanfear, R. M. López and S. K. Suslov, Exact wave functions for generalized harmonic oscillators, J. Russ. Laser Res. 32 (2011), no. 4, 352-361.
  • [50] R. M. López, S. K. Suslov, and J. M. Vega-Guzmán, Reconstructing the Schrödinger groups, Phys. Scr. 87 (2013), no. 3, 038118 (6 pages).
  • [51] Raquel M. López, Sergei K. Suslov, and José M. Vega-Guzmán, On a hidden symmetry of quantum harmonic oscillators, J. Difference Equ. Appl. 19 (2013), no. 4, 543-554. MR 3040814, https://doi.org/10.1080/10236198.2012.658384
  • [52] A. Mahalov, E. Suazo, and S. K. Suslov, Spiral laser beams in inhomogeneous media, Opt. Lett. 38 (2013), no. 15, 1-4.
  • [53] A. Mahalov and S. K. Suslov, An ``Airy gun'': Self-accelerating solutions of the time-dependent Schrödinger equation in vacuum, Phys. Lett. A 377 (2012), 33-38.
  • [54] A. Mahalov and S. K. Suslov, Wigner function approach to oscillating solutions of the $ 1D$-quintic nonlinear Schrödinger equation, J. Nonlinear Opt. Phys. & Mat. 22 (2013), no. 2, 1350013 (14 pages).
  • [55] Philippe Marcq, Hugues Chaté, and Robert Conte, Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation, Phys. D 73 (1994), no. 4, 305-317. MR 1280881 (95b:35036), https://doi.org/10.1016/0167-2789(94)90102-3
  • [56] Koji Mio, Tatsuki Ogino, Kazuo Minami, and Susumu Takeda, Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan 41 (1976), no. 1, 265-271. MR 0462141 (57 #2116)
  • [57] Koji Mio, Tatsuki Ogino, Kazuo Minami, and Susumu Takeda, Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan 41 (1976), no. 1, 265-271. MR 0462141 (57 #2116)
  • [58] Micheline Musette and Robert Conte, Analytic solitary waves of nonintegrable equations, Phys. D 181 (2003), no. 1-2, 70-79. MR 2003796 (2004f:35148), https://doi.org/10.1016/S0167-2789(03)00069-1
  • [59] A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991. Translated from the Russian. MR 1149380 (92m:33019)
  • [60] S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons, The inverse scattering method; Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. Translated from the Russian. MR 779467 (86k:35142)
  • [61] B. Øksendal, Stochastic differential equations, Springer-Verlag, Berlin, 2000.
  • [62] S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskiĭ, Principles of statistical radiophysics. 3, Elements of random fields; Springer-Verlag, Berlin, 1989. Translated from the second Russian edition by Alexander P. Repyev [A. P. Repev]. MR 1002949 (90h:85005)
  • [63] W. van Saarloos, Front propagation into unstable states, Phys. Rep. 386 (2003), 29-222.
  • [64] Wim van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Phys. D 56 (1992), no. 4, 303-367. MR 1169610 (93h:35195), https://doi.org/10.1016/0167-2789(92)90175-M
  • [65] B. Sanborn, S. K. Suslov, and L. Vinet, Dynamic invariants and Berry's phase for generalized driven harmonic oscillators, J. Russ. Laser Res. 32 (2011), no. 5, 486-494.
  • [66] G. A. Siviloglou and D. N. Christodoulides, Accelerating finite energy Airy beams, Opt. Lett. 32 (2007), no. 2, 979-981.
  • [67] G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Observation of accelerating Airy beams, Phys. Rev. Lett. 99 (2007), 213901 (4 pages).
  • [68] Ronald Smith, Giant waves, J. Fluid Mech. 77 (1976), no. 3, 417-431. MR 0436757 (55 #9696)
  • [69] E. Suazo and S. K. Suslov, Soliton-like solutions for nonlinear Schrödinger equation with variable quadratic Hamiltonians, J. Russ. Laser Res. 33 (2012), no. 1, 63-82.
  • [70] Sergei K. Suslov, On integrability of nonautonomous nonlinear Schrödinger equations, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3067-3082. MR 2917080, https://doi.org/10.1090/S0002-9939-2011-11176-6
  • [71] Masayoshi Tajiri, Similarity reductions of the one- and two-dimensional nonlinear Schrödinger equations, J. Phys. Soc. Japan 52 (1983), no. 6, 1908-1917. MR 710726 (84h:35154), https://doi.org/10.1143/ JPSJ.52.1908
  • [72] W. Tang and A. Mahalov, Stochastic Lagrangian dynamics for charged flows in the $ \emph {E-F}$ regions of ionosphere, Physics of Plasmas 203 (2013), 032305 (11 pages).
  • [73] V. I. Talanov, Focusing of light in cubic media, JETP Lett. 11 (1970), 199-201.
  • [74] Terence Tao, A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math. 15 (2009), 265-282. MR 2530148 (2010h:35379)
  • [75] Sergey Yu. Vernov, Construction of special solutions for nonintegrable systems, J. Nonlinear Math. Phys. 13 (2006), no. 1, 50-63. MR 2217118 (2006m:34222), https://doi.org/10.2991/jnmp.2006.13.1.5
  • [76] S. Yu. Vernov, Elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation, J. Phys. A 40 (2007), no. 32, 9833-9844. MR 2370547 (2008k:35456), https://doi.org/10.1088/1751-8113/40/32/009
  • [77] M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of waves, Nauka, Moscow, 1979 [in Russian]. MR 1127872 (92j:00028)
  • [78] S. N. Vlasov and V. I. Talanov, The parabolic equation in wave propagation theory (on the 50th anniversary of the first publication), Izv. Vyssh. Uchebn. Zaved. Radiofiz. 38 (1995), no. 1-2, 3-19 (Russian, with English and Russian summaries); English transl., Radiophys. and Quantum Electronics 38 (1995), no. 1-2, 1-12 (1996). MR 1427164 (97j:78019), https://doi.org/10.1007/BF01051853
  • [79] Miki Wadati, Heiji Sanuki, Kimiaki Konno, and Yoshi-Hiko Ichikawa, Circular polarized nonlinear Alfvén waves--a new type of nonlinear evolution equation in plasma physics, Conference on the Theory and Application of Solitons (Tucson, Ariz., 1976). Rocky Mountain J. Math. 8 (1978), no. 1-2, 323-331. MR 0496220 (58 #14796)
  • [80] Miki Wadati and Kiyoshi Sogo, Gauge transformations in soliton theory, J. Phys. Soc. Japan 52 (1983), no. 2, 394-398. MR 700302 (84g:35159), https://doi.org/10.1143/JPSJ.52.394
  • [81] 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. MR 0406174 (53 #9966)

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Additional Information

Alex Mahalov
Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287–1804
Email: mahalov@asu.edu

Sergei K. Suslov
Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287–1804
Email: sks@asu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12295-7
Keywords: Paraxial wave equations, Green's function, Airy--Gaussian--Hermite beams, Gaussian--Hermite beams, nonlinear Schr\"odinger equations, complex Ginsburg--Landau equations.
Received by editor(s): March 26, 2013
Published electronically: October 28, 2014
Additional Notes: This research was partially supported by AFOSR grant FA9550-11-1-0220.
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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