Publications of Stephanos Venakides

  1. Komineas, S; Melcher, C; Venakides, S, Chiral skyrmions of large radius, Physica D: Nonlinear Phenomena, vol. 418 (April, 2021), Elsevier
  2. Komineas, S; Melcher, C; Venakides, S, The profile of chiral skyrmions of small radius, Nonlinearity, vol. 33 no. 7 (July, 2020), pp. 3395-3408, London Mathematical Society
  3. Komineas, S; Melcher, C; Venakides, S, Traveling domain walls in chiral ferromagnets, Nonlinearity, vol. 32 no. 7 (May, 2019), pp. 2392-2412, London Mathematical Society
  4. Pérez-Arancibia, C; Shipman, SP; Turc, C; Venakides, S, Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies, Communications in Computational Physics, vol. 26 no. 1 (January, 2019), pp. 265-310, Global Science Press
  5. Aristotelous, AC; Crawford, JM; Edwards, GS; Kiehart, DP; Venakides, S, Mathematical models of dorsal closure., Progress in biophysics and molecular biology, vol. 137 (September, 2018), pp. 111-131
  6. Perez-Arancibia, C; Shipman, S; Turc, C; Venakides, S, DDM solutions of quasiperiodic transmission problems in layered media via robust boundary integral equations at all frequencies, Communications in Computational Physics (May, 2018), Global Science Press
  7. Bruno, OP; Shipman, SP; Turc, C; Venakides, S, Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies, Proc. R. Soc. A 2017 473 20170242, vol. 473 no. 2207 (November, 2017), pp. 20170242, The Royal Society
  8. Kiehart, DP; Crawford, JM; Aristotelous, A; Venakides, S; Edwards, GS, Cell Sheet Morphogenesis: Dorsal Closure in Drosophila melanogaster as a Model System., Annual review of cell and developmental biology, vol. 33 (October, 2017), pp. 169-202
  9. Bruno, OP; Shipman, SP; Turc, C; Venakides, S, Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 472 no. 2191 (July, 2016), pp. 20160255
  10. Komineas, S; Shipman, SP; Venakides, S, Lossless polariton solitons, Physica D: Nonlinear Phenomena, vol. 316 (February, 2016), pp. 43-56, Elsevier BV
  11. Sergey Belov and Stephanos Venakides, Smooth parametric dependence of asymptotics of the semiclassical focusing NLS, Analysis & PDE, vol. 8 no. 2 (April, 2015), pp. 257-288
  12. Komineas, S; Shipman, SP; Venakides, S, Continuous and discontinuous dark solitons in polariton condensates, Physical Review B - Condensed Matter and Materials Physics, vol. 91 no. 13 (April, 2015), American Physical Society (APS), ISSN 1098-0121
  13. Belov, S; Venakides, S, Smooth parametric dependence of asymptotics of the semiclassical focusing NLS, Analysis and PDE, vol. 8 no. 2 (January, 2015), pp. 257-288, Mathematical Sciences Publishers
  14. Belov, S; Venakides, S, Long-time limit studies of an obstruction in the g-function mechanism for semiclassical focusing NLS (2015)
  15. Stavros Komineas, Stephen P. Shipman, Stephanos Venakides, Lossless Polariton Solitons, arXiv (2014)
  16. Oscar P. Bruno, Stephen P. Shipman, Catalin Turc, Stephanos Venakides, Efficient Evaluation of Doubly Periodic Green Functions in 3D Scattering, Including Wood Anomaly Frequencies, ArXiv>Mathematics > Analysis of PDEs (July 4, 2013)
  17. Jackson, AD; Huang, D; Gauthier, DJ; Venakides, S, Destructive impact of imperfect beam collimation in extraordinary optical transmission., Journal of the Optical Society of America. A, Optics, image science, and vision, vol. 30 no. 6 (June, 2013), pp. 1281-1290, ISSN 1084-7529 (doi: 10.1364/JOSAA.30.001281..)
  18. Shipman, SP; Venakides, S, An exactly solvable model for nonlinear resonant scattering, Nonlinearity, vol. 25 no. 9 (September, 2012), pp. 2473-2501, IOP Publishing, ISSN 0951-7715 (doi:10.1088/0951-7715/25/9/2473.)
  19. Tovbis, A; Venakides, S, Semiclassical limit of the scattering transform for the focusing nonlinear Schrödinger equation, International Mathematics Research Notices, vol. 2012 no. 10 (May, 2012), pp. 2212-2271, Oxford University Press (OUP), ISSN 1073-7928 (doi:10.1093/imrn/rnr092.)
  20. Belov, S; Venakides, S, Perturbation of Riemann-Hilbert jump contours: smooth parametric dependence with application to semiclassical focusing NLS (August, 2011)
  21. Tovbis, A; Venakides, S, Nonlinear steepest descent asymptotics for semiclassical limit of Integrable systems: Continuation in the parameter space, Communications in Mathematical Physics, vol. 295 no. 1 (February, 2010), pp. 139-160, Springer Nature, ISSN 0010-3616
  22. Layton, AT; Toyama, Y; Yang, G-Q; Edwards, GS; Kiehart, DP; Venakides, S, Drosophila morphogenesis: tissue force laws and the modeling of dorsal closure., HFSP journal, vol. 3 no. 6 (December, 2009), pp. 441-460, HFSP
  23. Lefew, WR; Venakides, S; Gauthier, DJ, Accurate description of optical precursors and their relation to weak-field coherent optical transients, Physical Review A - Atomic, Molecular, and Optical Physics, vol. 79 no. 6 (June, 2009), pp. 063842, American Physical Society (APS), ISSN 1050-2947
  24. Tovbis, A; Venakides, S, Determinant form of the complex phase function of the steepest descent analysis of Riemann-Hilbert problems and its application to the focusing nonlinear schrödinger equation, International Mathematics Research Notices, vol. 2009 no. 11 (February, 2009), pp. 2056-2080, Oxford University Press (OUP), ISSN 1073-7928
  25. Tovbis, A; Venakides, S, Determinant form of modulation equations for the semiclassical focusing Nonlinear Schr\" odinger equation (2009)
  26. Ptitsyna, N; Shipman, SP; Venakides, S, Fano resonance of waves in periodic slabs, Mathematical Methods in Electromagnetic Theory, MMET, Conference Proceedings (September, 2008), pp. 73-78
  27. Tovbis, A; Venakides, S; Zhou, X, Semiclassical Focusing Nonlinear Schrodinger equation in the pure radiation case: Riemann-Hilbert Problem approach, edited by Baik, J; Kriecherbauer, T; Li, LC; McLaughlin, KDT; Tomei, C, Integrable Systems and Random Matrices: In Honor of Percy Deift, vol. 458 (2008), pp. 117-144, AMER MATHEMATICAL SOC, ISBN 978-0-8218-4240-9
  28. Tovbis, A; Venakides, S; Zhou, X, Semiclassical focusing nonlinear schrödinger equation i: Inverse scattering map and its evolution for radiative initial data, International Mathematics Research Notices, vol. 2007 no. Article ID rnm094, 54 pages. doi:10. (December, 2007), Oxford University Press (OUP), ISSN 1073-7928
  29. Buckingham, R; Venakides, S, Long-time asymptotics of the nonlinear Schrödinger equation shock problem, Communications on Pure and Applied Mathematics, vol. 60 no. 9 (September, 2007), pp. 1349-1414, WILEY, ISSN 0010-3640
  30. Peralta, XG; Toyama, Y; Hutson, MS; Montague, R; Venakides, S; Kiehart and, DP; Edwards, GS, Resiliency, coordination, and synchronization of dorsal closure during Drosophila morphogenesis, Biophysical Journal, vol. 92 no. 7 (April, 2007), pp. 2583-2596, ISSN 0006-3495
  31. Peralta, XG; Toyama, Y; Hutson, MS; Montague, R; Venakides, S; Kiehart, DP; Edwards, GS, Upregulation of forces and morphogenic asymmetries in dorsal closure during Drosophila development., Biophysical journal, vol. 92 no. 7 (April, 2007), pp. 2583-2596, ISSN 0006-3495
  32. Buckingham, R; Tovbis, A; Venakides, S; Zhou, X, The semiclassical focusing nonlinear Schrodinger equation, in "Recent Advances in Nonlinear Partial Differentila Equations and Applications'', Proceedings of Symposia in Applied Mathematics, edited by Bonilla, LL; Carpio, A; Vega, JM; Venakides, S, Recent Advances in Nonlinear Partial Differential Equations and Applications, vol. 65 (2007), pp. 47-80, AMER MATHEMATICAL SOC, ISBN 978-0-8218-4211-9
  33. Tovbis, A; Venakides, S; Zhou, X, On the long-time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation: Pure radiation case, Communications on Pure and Applied Mathematics, vol. 59 no. 10 (January, 2006), pp. 1379-1432, WILEY, ISSN 0010-3640
  34. Shipman, SP; Venakides, S, Resonant transmission near nonrobust periodic slab modes., Physical review. E, Statistical, nonlinear, and soft matter physics, vol. 71 no. 2 Pt 2 (February, 2005), pp. 026611, ISSN 1539-3755
  35. Peralta, XG; Toyama, Y; Wells, A; Tokutake, Y; Hutson, MS; Venakides, S; Kiehart, DP; Edwards, GS, Force regulation during dorsal closure in Drosophila, Molecular Biology of the Cell, vol. 15 (November, 2004), pp. 403A-403A, American Society for Cell Biology
  36. Tovbis, A; Venakides, S; Zhou, X, On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation, Communications on Pure and Applied Mathematics, vol. 57 no. 7 (July, 2004), pp. 877-985, WILEY, ISSN 0010-3640
  37. Shipman, SP; Venakides, S, Resonance and bound states in photonic crystal slabs, SIAM Journal on Applied Mathematics, vol. 64 no. 1 (October, 2003), pp. 322-342, Society for Industrial & Applied Mathematics (SIAM), ISSN 0036-1399
  38. Hutson, MS; Tokutake, Y; Chang, M-S; Bloor, JW; Venakides, S; Kiehart, DP; Edwards, GS, Forces for morphogenesis investigated with laser microsurgery and quantitative modeling., Science (New York, N.Y.), vol. 300 no. 5616 (April, 2003), pp. 145-149
  39. Lipton, RP; Shipman, SP; Venakides, S, Optimization of Resonances in Photonic Crystal Slabs, Proceedings of SPIE - The International Society for Optical Engineering, vol. 5184 (January, 2003), pp. 168-177, SPIE
  40. Hutson, S; Tokutake, Y; Chang, M; Bloor, JW; Venakides, S; Kiehart, DP; Edwards, GS, Measuring the forces that drive morphogenesis: Laser-microsurgery and quantitative modeling applied to dorsal closure in Drosophila, Molecular Biology of the Cell, vol. 13 (November, 2002), pp. 476A-476A, American Society for Cell Biology
  41. Haider, MA; Shipman, SP; Venakides, S, Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances, SIAM Journal on Applied Mathematics, vol. 62 no. 6 (July, 2002), pp. 2129-2148, Society for Industrial & Applied Mathematics (SIAM)
  42. El, GA; Krylov, AL; Venakides, S, Unified approach to KdV modulations, Communications on Pure and Applied Mathematics, vol. 54 no. 10 (October, 2001), pp. 1243-1270, WILEY
  43. Deift, P; Kriecherbauer, T; McLaughlin, KR; Venakides, S; Zhou, X, A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials, Journal of Computational and Applied Mathematics, vol. 133 no. 1-2 (August, 2001), pp. 47-63, Elsevier BV, ISSN 0377-0427
  44. El, GA; Krylov, AL; Molchanov, SA; Venakides, S, Soliton turbulence as a thermodynamic limit of stochastic soliton lattices, Physica D: Nonlinear Phenomena, vol. 152-153 (May, 2001), pp. 653-664, Elsevier BV
  45. Georgieva, A; Kriecherbauer, T; Venakides, S, 1:2 resonance mediated second harmonic generation in a 1-D nonlinear discrete periodic medium, SIAM Journal on Applied Mathematics, vol. 61 no. 5 (January, 2001), pp. 1802-1815, Society for Industrial & Applied Mathematics (SIAM)
  46. Tovbis, A; Venakides, S, The eigenvalue problem for the focusing nonlinear Schrödinger equation: New solvable cases, Physica D: Nonlinear Phenomena, vol. 146 no. 1-4 (November, 2000), pp. 150-164, Elsevier BV
  47. Venakides, S; Haider, MA; Papanicolaou, V, Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures, SIAM Journal on Applied Mathematics, vol. 60 no. 5 (January, 2000), pp. 1686-1706, Society for Industrial & Applied Mathematics (SIAM)
  48. Reed, D; Venakides, S, Studying the asymptotics of Selberg-type integrals, edited by Spigler, R, Applied and Industrial Mathematics, Venice-2, 1998 (2000), pp. 187-198, SPRINGER, ISBN 0-7923-6152-0
  49. Beaky, MM; Burk, JB; Everitt, HO; Haider, MA; Venakides, S, Two-dimensional photonic crystal fabry-perot resonators with lossy dielectrics, IEEE Transactions on Microwave Theory and Techniques, vol. 47 no. 11 (December, 1999), pp. 2085-2091, Institute of Electrical and Electronics Engineers (IEEE), ISSN 0018-9480
  50. Filip, AM; Venakides, S, Existence and modulation of traveling waves in particle chains, Communications on Pure and Applied Mathematics, vol. 52 no. 6 (January, 1999), pp. 693-735
  51. Cheng, PJ; Venakides, S; Zhou, X, Long-time asymptotics for the pure radiation solution of the sine-Gordon equation, Communications in Partial Differential Equations, vol. 24 no. 7-8 (January, 1999), pp. 1195-1262, Informa UK Limited
  52. Georgieva, A; Kriecherbauer, T; Venakides, S, Wave propagation and resonance in a one-dimensional nonlinear discrete periodic medium, SIAM Journal on Applied Mathematics, vol. 60 no. 1 (January, 1999), pp. 272-294, Society for Industrial & Applied Mathematics (SIAM)
  53. Deift, P; Kriecherbauer, T; McLaughlin, KTR; Venakides, S; Zhou, X, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Communications on Pure and Applied Mathematics, vol. 52 no. 11 (January, 1999), pp. 1335-1425, WILEY
  54. Deift, P; Kriecherbauer, T; Mclaughlin, KTR; Venakides, S; Zhou, X, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Communications on Pure and Applied Mathematics, vol. 52 no. 12 (January, 1999), pp. 1491-1552
  55. Deift, P; Venakides, S; Zhou, X, An extension of the steepest descent method for Riemann-Hilbert problems: the small dispersion limit of the Korteweg-de Vries (KdV) equation., Proceedings of the National Academy of Sciences of the United States of America, vol. 95 no. 2 (January, 1998), pp. 450-454, ISSN 0027-8424
  56. McDonald, MA; Venakides, S, Renormalization of the τ-functions for integrable systems: A model problem, Communications on Pure and Applied Mathematics, vol. 51 no. 8 (January, 1998), pp. 937-966, WILEY
  57. Deift, P; Kriecherbauer, T; McLaughlin, KTR; Venakides, S; Zhou, X, Asymptotics for Polynomials Orthogonal with Respect to Varying Exponential Weights, International Mathematics Research Notices no. 16 (December, 1997), pp. X-782
  58. Deift, P; Venakides, S; Zhou, X, New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann-Hilbert Problems, International Mathematics Research Notices no. 6 (December, 1997), pp. 284-299
  59. Bonilla, LL; Kindelan, M; Moscoso, M; Venakides, S, Periodic generation and propagation of traveling fronts in dc voltage biased semiconductor superlattices, SIAM Journal on Applied Mathematics, vol. 57 no. 6 (January, 1997), pp. 1588-1614, Society for Industrial & Applied Mathematics (SIAM)
  60. Deift, P; Venakides, S; Zhou, X, New results in small dispersion kdV by an extension of the steepest descent method for Riemann-Hilbert problems, International Mathematics Research Notices no. 6 (1997), pp. 285-299, Oxford University Press (OUP): Policy B - Oxford Open Option A
  61. Deift, P; Kriecherbauer, T; McLaughlin, KTR; Venakides, S; Zhou, X, Asymptotics for polynomials orthogonal with respect to varying exponential weights, International Mathematics Research Notices no. 16 (1997), pp. 759-782, Oxford University Press (OUP): Policy B - Oxford Open Option A
  62. Deift, P; Kriecherbauer, T; Venakides, S, Forced lattice vibrations: Part I, Communications on Pure and Applied Mathematics, vol. 48 no. 11 (January, 1995), pp. 1187-1249, WILEY
  63. Deift, P; Kriecherbauer, T; Venakides, S, Forced lattice vibrations: Part II, Communications on Pure and Applied Mathematics, vol. 48 no. 11 (January, 1995), pp. 1251-1298, WILEY
  64. Deift, P; Kriecherbauer, T; Venakides, S, Forced Lattice Vibrations -- A Videotext (September, 1994)
  65. Bonilla, LL; Higuera, FJ; Venakides, S, Gunn effect: Instability of the steady state and stability of the solitary wave in long extrinsic semiconductors, SIAM Journal on Applied Mathematics, vol. 54 no. 6 (January, 1994), pp. 1521-1541, Society for Industrial & Applied Mathematics (SIAM)
  66. Deift, P; Venakides, S; Zhou, X, The collisionless shock region for the long‐time behavior of solutions of the KdV equation, Communications on Pure and Applied Mathematics, vol. 47 no. 2 (January, 1994), pp. 199-206, WILEY
  67. Zhang, T; Venakides, S, Periodic limit of inverse scattering, Communications on Pure and Applied Mathematics, vol. 46 no. 6 (January, 1993), pp. 819-865, WILEY
  68. Venakides, S; Deift, P; Oba, R, The toda shock problem, Communications on Pure and Applied Mathematics, vol. 44 no. 8-9 (January, 1991), pp. 1171-1242, WILEY
  69. Venakides, S, The Korteweg-Devries Equation with Small Dispersion - Higher-Order Lax Levermore Theory, edited by SPIGLER, R, Journal of Applied and Industrial Mathematics, vol. 56 (1991), pp. 255-262, KLUWER ACADEMIC PUBL, ISBN 0-7923-0521-3
  70. Reed, MC; Venakides, S; Blum, JJ, Approximate traveling waves in linear reaction-hyperbolic equations, SIAM Journal on Applied Mathematics, vol. 50 no. 1 (January, 1990), pp. 167-180, Society for Industrial & Applied Mathematics (SIAM)
  71. Venakides, S, The korteweg‐de vries equation with small dispersion: Higher order lax‐levermore theory, Communications on Pure and Applied Mathematics, vol. 43 no. 3 (January, 1990), pp. 335-361, WILEY
  72. Venakides, S, The continuum limit of theta functions, Communications on Pure and Applied Mathematics, vol. 42 no. 6 (January, 1989), pp. 711-728, WILEY
  73. Venakides, S, The Small Dispersion Limit of the Korteweg-Devries Equation, edited by DAFERMOS, CM; LADAS, G; PAPANICOLAOU, G, Differential Equations, vol. 118 (1989), pp. 725-737, Marcel Dekker, ISBN 0-8247-8077-9
  74. Venakides, S, The infinite period limit of the inverse formalism for periodic potentials, Communications on Pure and Applied Mathematics, vol. 41 no. 1 (January, 1988), pp. 3-17, WILEY
  75. Venakides, S, The Zero Dispersion Limit of the Korteweg-Devries Equation with Periodic Initial Data, Transactions of the American Mathematical Society, vol. 301 no. 1 (May, 1987), pp. 189-226, American Mathematical Society
  76. Venakides, S, The zero dispersion limit of the korteweg-de vries equation with periodic initial data, Transactions of the American Mathematical Society, vol. 301 no. 1 (January, 1987), pp. 189-226, American Mathematical Society (AMS)
  77. Venakides, S, Long time asymptotics of the korteweg-de vries equation, Transactions of the American Mathematical Society, vol. 293 no. 1 (January, 1986), pp. 411-419, American Mathematical Society (AMS)
  78. Venakides, S, Long-Time Asymptotics of the Korteweg-Devries Equation, Transcations of the American Mathematical Society, vol. 293 no. 1 (January, 1986), pp. 411-419, JSTOR
  79. Venakides, S, The generation of modulated wavetrains in the solution of the Korteweg—de vries equation, Communications on Pure and Applied Mathematics, vol. 38 no. 6 (January, 1985), pp. 883-909, WILEY
  80. Venakides, S, The zero dispersion limit of the korteweg‐de vries equation for initial potentials with non‐trivial reflection coefficient, Communications on Pure and Applied Mathematics, vol. 38 no. 2 (January, 1985), pp. 125-155, WILEY
  81. El, G.A.; Krylov, A.L.; Molchanov, S.A.; Venakides, S., Soliton turbulence as a thermodynamic limit of stochastic soliton lattices. In Advances in nonlinear mathematics and science., Physica D 152/153 (2001), 653--664
  82. S. Venakides, M. Haider, V. Papanicolaou, Boundary Integral Calculations of 2-d Electromagnetic Scattering by Photonic Crystal Fabry-Perot Structures, SIAM J. Appl. Math. vol. 60/5, (2000), pp. 1636-1706
  83. A. Georgieva, T. Kriecherbauer, Stephanos Venakides, Wave Propagation and Resonance in a 1-d Nonlinear Discrete Periodic Medium, SIAM J. Appl. Math., vol. 60/1, (1999), pp. 272-294
  84. P. Deift, T. Kriecherbauer, K. T-R McLaughlin,S. Venakides, X. Zhou, Strong Asymptotics of Orhtogonal Polynomials with Respect to Exponential Weights, CPAM, vol.52 (1999) 1491-1552.
  85. M. McDonald, S. Venakides, Renormalization of the Tau Function for Integrable Systems: A Model Problem, CPAM, Vol 51, 1998, 937-966.
  86. P. Deift, S. Venakides, X. Zhou, An Extension of the Method of Steepest Descent for Riemann-Hilbert Problems: The Small Dispersion Limit of the Korteweg-de Vries (KdV) Equation, Proc. Ntl. Acad. Sc. USA, vol. 95, Jan 1998, 450-454.
  87. P. Deift, S. Venakides, X. Zhou, New Results in the Small-Dispersion KdV by an Extension of the Method of Steepest Descent for Riemann-Hilbert Problems, IMRN, 1997, N0. 6, 285-299.
  88. P. Deift, T. Kriecherbauer, K. T-R McLaughlin,S. Venakides, X. Zhou, Asymptotics of Polynomials Orthogonal with Respect to Varying Exponential Weights, IMRN, 1997 No 16, pp. 759-782
  89. P. Deift, T. Kriecherbauer, S. Venakides, Forced Lattice Vibrations Part II, Comm. Pure Appl. Math. 48, 1995, 1251-1298.
  90. P. Deift, T. Kriecherbauer, S. Venakides, Forced Lattice Vibrations Part I, Comm. Pure Appl. Math. 48,1995, 1187-1250.
  91. L. L. Bonilla, F. Higuera, S. Venakides, The Stability of the Steady State of the Gunn Oscillator, SIAM J. Appl. Math. vol. 54, No 6, (1994), pp. 1521-1541.
  92. P. Deift, S. Venakides, X. Zhou, The Collisionless Shock Region for the Long Time Behavior of the Solutions of the KdV Equation, CPAM. vol. 47, (1994), pp. 199-206.
  93. P. D. Lax, C. D. Levermore, S. Venakides, The Generation and Propagation of Oscillations in Dispersive IVP's and their Limiting Behavior, Important Developments in Soliton Theory 1980--1990}, T. Fokas and V.E. Zakharov eds., Springer-Verlag, Berlin (1992).
  94. S. Venakides, The solution of completely integrable systems in the continuum limit of the spectral data, IMA Proceedings, vol. 2, (1986) pp. 337-356..
  95. S. Venakides, The zero-dispersion limit of the Korteweg-de Vries equation with non-trivial reflection coefficient, Comm. Pure and Appl. Math. 38, pp. 125-155, 1985.