×

Some special solutions to the hyperbolic NLS equation. (English) Zbl 1459.76025

Summary: The Hyperbolic Nonlinear Schrödinger equation (HypNLS) arises as a model for the dynamics of three-dimensional narrow-band deep water gravity waves. In this study, the symmetries and conservation laws of this equation are computed. The Petviashvili method is then exploited to numerically compute bi-periodic time-harmonic solutions of the HypNLS equation. In physical space they represent non-localized standing waves. Non-trivial spatial patterns are revealed and an attempt is made to describe them using symbolic dynamics and the language of substitutions. Finally, the dynamics of a slightly perturbed standing wave is numerically investigated by means a highly accurate Fourier solver.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
76M99 Basic methods in fluid mechanics

Software:

GeM; FFTW; Matlab; advanpix
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ablowitz, M. J.; Segur, H., On the evolution of packets of water waves, J Fluid Mech, 92, 691-715 (1979) · Zbl 0413.76009
[2] Ablowitz, M. J.; Segur, H., Solitons and the Inverse Scattering Transform (1981), Soc Ind Appl Math · Zbl 0472.35002
[3] Allouche, J.-P.; Shallit, J. O., Automatic sequences - theory, applications, generalizations (2003), Cambridge University Press · Zbl 1086.11015
[4] Álvarez, J.; Durán, A., Petviashvili type methods for traveling wave computations: I. Analysis of convergence, J Comp Appl Math, 266, 39-51 (2014) · Zbl 1293.65079
[5] Arnoux, P.; Rauzy, G., Représentation géométrique de suites de complexité \(2n+1\), Bull Soc Math France, 119, 2, 199-215 (1991) · Zbl 0789.28011
[6] Benney, D. J.; Newell, A. C., The propagation of nonlinear wave envelopes, J Math and Physics, 46, 133-139 (1967) · Zbl 0153.30301
[7] Berthé, V.; Vuillon, L., Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences, Discrete Math, 223, 1-3, 27-53 (2000) · Zbl 0970.68124
[8] Boyd, J. P., Chebyshev and Fourier spectral methods, New York (2000)
[9] Chen, M.; Iooss, G., Standing waves for a two-way model system for water waves, Eur J Mech - B/Fluids, 24, 1, 113-124 (2005) · Zbl 1060.76018
[10] Chen, M.; Iooss, G., Periodic wave patterns of two-dimensional Boussinesq systems, Eur J Mech - B/Fluids, 25, 4, 393-405 (2006) · Zbl 1122.76018
[11] Chen, M.; Iooss, G., Asymmetric periodic traveling wave patterns of two-dimensional Boussinesq systems, Physica D, 237, 10-12, 1539-1552 (2008) · Zbl 1143.76396
[12] Cheviakov, A. F., GeM software package for computation of symmetries and conservation laws of differential equations, Comp Phys Comm, 176, 1, 48-61 (2007) · Zbl 1196.34045
[13] Chu, V. H.; Mei, C. C., On slowly-varying Stokes waves, J Fluid Mech, 41, 873-887 (1970) · Zbl 0221.76005
[14] Clamond, D.; Grue, J., A fast method for fully nonlinear water-wave computations, J Fluid Mech, 447, 337-355 (2001) · Zbl 0997.76064
[15] Conti, C.; Trillo, S.; Di Trapani, P.; Valiulis, G.; Piskarskas, A.; Jedrkiewicz, O., Nonlinear electromagnetic X waves, Phys Rev Lett, 90, 17 (2003)
[16] Craig, W.; Nicholls, D. P., Traveling gravity water waves in two and three dimensions, Eur J Mech - B/Fluids, 21, 6, 615-641 (2002) · Zbl 1084.76509
[17] Davey, A.; Stewartson, K., On three-dimensional packets of surface waves, Proc R Soc A, 338, 1613, 101-110 (1974) · Zbl 0282.76008
[18] de Bruijn, N. G., Algebraic theory of Penrose’s nonperiodic tilings of the plane. I, II, Nederl Akad Wetensch Indag Math, 43, 1, 39-66 (1981) · Zbl 0457.05021
[19] Drazin, P. G.; Johnson, R. S., Solitons: an introduction (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0661.35001
[20] Dysthe, K. B., Note on a modification to the nonlinear Schrödinger equation for application to deep water, Proc R Soc Lond A, 369, 105-114 (1979) · Zbl 0429.76014
[21] Ebadi, G.; Biswas, A., The G’/G method and 1-soliton solution of the Davey-Stewartson equation, Math Comput Model, 53, 5-6, 694-698 (2011) · Zbl 1217.35171
[22] Ebadi, G.; Krishnan, E. V.; Labidi, M.; Zerrad, E.; Biswas, A., Analytical and numerical solutions to the Davey-Stewartson equation with power-law nonlinearity, Waves Random Complex Media, 21, 4, 559-590 (2011) · Zbl 1274.76167
[23] Fermi, E.; Pasta, J.; Ulam, S., Studies of nonlinear problems, Technical Report (1955), Los Alamos National Laboratory: Los Alamos National Laboratory Los Alamos, USA
[24] for MATLAB, M. C.T., v4.3.3.12185 (2017), Advanpix LLC.: Advanpix LLC. Tokyo, Japan
[25] Frigo, M.; Johnson, S. G., The design and implementation of FFTW3, Proc IEEE, 93, 2, 216-231 (2005)
[26] Frougny, C.; Vuillon, L., Coding of two-dimensional constraints of finite type by substitutions, J Automata Languages Combinatorics, 10, 4, 465-482 (2005) · Zbl 1153.68401
[27] Fructus, D.; Clamond, D.; Kristiansen, O.; Grue, J., An efficient model for threedimensional surface wave simulations. Part I: free space problems, J Comput Phys, 205, 665-685 (2005) · Zbl 1087.76016
[28] Galaktionov, V. A., Geometric Sturmian theory of nonlinear parabolic equations and applications (2004), Chapman and Hall/CRC: Chapman and Hall/CRC Boca Raton, London, New York, Washington, DC · Zbl 1075.35017
[29] Ghanem, R.; Spanos, P., Stochastic finite elements: a spectral approach (2003), Dover Publications Inc.: Dover Publications Inc. Mineola N.Y.
[30] Ghidaglia, J.-M.; Saut, J.-C., On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3, 2, 475-506 (1990) · Zbl 0727.35111
[31] Ghidaglia, J.-M.; Saut, J.-C., Nonelliptic Schrödinger equations, J Nonlinear Sci, 3, 1, 169-195 (1993) · Zbl 0808.35135
[32] Ghidaglia, J.-M.; Saut, J.-C., Nonexistence of travelling wave solutions to nonelliptic nonlinear Schrödinger equations, J Nonlinear Sci, 6, 2, 139-145 (1996) · Zbl 0848.35123
[33] Hasimoto, H.; Ono, H., Nonlinear modulation of gravity waves, J Phys Soc Jpn, 33, 3, 805-811 (1972)
[34] Henderson, D.; Segur, H.; Carter, J. D., Experimental evidence of stable wave patterns on deep water, J Fluid Mech, 658, 247-278 (2010) · Zbl 1205.76012
[35] Iooss, G.; Plotnikov, P. I.; Toland, J. F., Standing waves on an infinitely deep perfect fluid under gravity, Arch Rat Mech Anal, 177(3), 367-478 (2005) · Zbl 1176.76017
[36] Jafari, H.; Sooraki, A.; Talebi, Y.; Biswas, A., The first integral method and traveling wave solutions to Davey-Stewartson equation, Nonlinear Anal, 17, 2, 182-193 (2012) · Zbl 1311.35289
[37] Johnson, R. S., A modern introduction to the mathematical theory of water waves (1997), Cambridge University Press, Cambridge · Zbl 0892.76001
[38] Kevrekidis, P.; Nahmod, A. R.; Zeng, C., Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D, Nonlinearity, 24, 5, 1523-1538 (2011) · Zbl 1216.35142
[39] Klein, C.; Roidot, K., Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations, SIAM J Sci Comput, 33, 6, 3333-3356 (2011) · Zbl 1298.65141
[40] Lahini, Y.; Frumker, E.; Silberberg, Y.; Droulias, S.; Hizanidis, K.; Morandotti, R., Discrete X-wave formation in nonlinear waveguide arrays, Phys Rev Lett, 98, 2 (2007)
[41] Lakoba, T. I.; Yang, J., A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity, J Comp Phys, 226, 1668-1692 (2007) · Zbl 1126.35052
[42] Lawson, J. D., Generalized Runge-Kutta processes for stable systems with large Lipschitz constants, SIAM J Numer Anal, 4, 3, 372-380 (1967) · Zbl 0223.65030
[43] Lind, D. A.; Marcus, B. H., An introduction to symbolic dynamics and coding (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1106.37301
[44] Lucia, D. J.; Beran, P. S.; Silva, W. A., Reduced-order modeling: new approaches for computational physics, Progress Aerospace Sci, 40, 1-2, 51-117 (2004)
[45] Milewski, P.; Tabak, E., A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows, SIAM J Sci Comput, 21(3), 1102-1114 (1999) · Zbl 0953.65073
[46] Morse, M.; Hedlund, G. A., Symbolic dynamics II. Sturmian trajectories, Am J Math, 62, 1-42 (1940) · Zbl 0022.34003
[47] Osborne, A., Nonlinear ocean waves and the inverse scattering transform, 97 (2010), Elsevier · Zbl 1250.86006
[48] Osborne, A. R.; Onorato, M.; Serio, M., The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains, Phys Lett A, 275, 5-6, 386-393 (2000) · Zbl 1115.76315
[49] Pelinovsky, D. E., A mysterious threshold for transverse instability of deep-water solitons, Math Comp Simul, 55, 4-6, 585-594 (2001) · Zbl 0987.76031
[50] Pelinovsky, D. E.; Stepanyants, Y. A., Convergence of Petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations, SIAM J Num Anal, 42, 1110-1127 (2004) · Zbl 1086.65098
[51] Pelinovsky, E. N.; Slunyaev, A. V.; Talipova, T.; Kharif, C., Nonlinear parabolic equation and extreme waves on the sea surface, Radiophys Quant Elect, 46, 7, 451-463 (2003)
[52] Petviashvili, V. I., Equation of an extraordinary soliton, Sov J Plasma Phys, 2(3), 469-472 (1976)
[53] Rowley, C. W., Model reduction for fluids, using balanced proper orthogonal decomposition, Int J Bifurcation Chaos, 15, 03, 997-1013 (2005) · Zbl 1140.76443
[54] Sen, A.; Karney, C. F.F.; Johnston, G. L.; Bers, A., Three-dimensional effects in the non-linear propagation of lower-hybrid waves, Nuclear Fus, 18, 2, 171-179 (1978)
[55] Senechal, M., Quasicrystals and geometry (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0828.52007
[56] Sirovich, L., Turbulence and the dynamics of coherent structures: Parts I-III, Q Appl Math, 45, 561-590 (1987) · Zbl 0676.76047
[57] Söderlind, G., Digital filters in adaptive time-stepping, ACM Trans Math Software, 29, 1-26 (2003) · Zbl 1097.93516
[58] Söderlind, G.; Wang, L., Adaptive time-stepping and computational stability, J Comp Appl Math, 185(2), 225-243 (2006) · Zbl 1077.65086
[59] Stoker, J. J., Water waves: the mathematical theory with applications (1957), Interscience: Interscience New York · Zbl 0078.40805
[60] Stokes, G. G., On the theory of oscillatory waves, Trans Camb Phil Soc, 8, 441-455 (1847)
[61] Sturm, C., Mémoire sur les équations différentielles linéaires du second ordre, J Math Pures Appl, 1, 106-186 (1836)
[62] Sulem, C.; Sulem, P.-L., The nonlinear Schrödinger equation. Self-focusing and wave collapse (1999), Springer-Verlag, New York · Zbl 0928.35157
[63] Trefethen, L. N., Spectral methods in MatLab (2000), Society for Industrial and Applied Mathematics, Philadelphia, PA, USA · Zbl 0953.68643
[64] Trulsen, K.; Dysthe, K. B., A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water, Wave Motion, 24, 281-289 (1996) · Zbl 0929.76022
[65] Verner, J. H., Explicit Runge-Kutta methods with estimates of the local truncation error, SIAM J Num Anal, 15(4), 772-790 (1978) · Zbl 0403.65029
[66] Vuillon, L., Balanced words, Bull Belgian Math Soc-Simon Stevin, 10, 5, 787-805 (2003) · Zbl 1070.68129
[67] Vuillon L., Dutykh D., Fedele F.. Animation of a perturbed pattern dynamics under the hyperbolic NLS equation. 2014. http://youtu.be/hIhLlsV69OQ; Vuillon L., Dutykh D., Fedele F.. Animation of a perturbed pattern dynamics under the hyperbolic NLS equation. 2014. http://youtu.be/hIhLlsV69OQ
[68] Yildirim, A.; Pagnaleh, A. S.; Mirzazadeh, M.; Moosaei, H.; Biswas, A., New exact traveling wave solutions for DS-I and DS-II equations, Nonlinear Anal, 17, 3, 369-378 (2012) · Zbl 1308.35052
[69] Yuen, H. C.; Lake, B. M., Nonlinear dynamics of deep-water gravity waves, Adv App Mech, 22, 67-229 (1982) · Zbl 0567.76026
[70] Zakharov, V. E., Stability of periodic waves of finite amplitude on the surface of a deep fluid, J Appl Mech Tech Phys, 9, 190-194 (1968)
[71] Zakharov, V. E.; Kuznetsova, E. A., Solitons and collapses: two evolution scenarios of nonlinear wave systems, Phys-Usp, 55, 6, 535-556 (2012)
[72] Zakharov, V. E.; Shabat, A. B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Phys-JETP, 34, 62-69 (1972)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.