×
Jiménez, S.; Pascual, P.; Aguierre, C.; Vázquez, L.

A panoramic view of some perturbed nonlinear wave equations. (English) Zbl 1063.65082

Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, No. 1, 1-40 (2004).
Summary: We present a panoramic view of numerical simulations associated with nonlinear wave equations which appear in different experimental contexts. Mainly, we deal with scalar wave equations, but also the Maxwell equations in nonlinear media are studied. A basic part of this work is devoted to the construction and verification of numerical schemes on a physical basis. The stochastic perturbations of scalar wave equations are especially analyzed by analytical and numerical approaches. Also, other kinds of perturbations are considered, like nonlocal ones. Finally, a summary of promising experimental results from the numerical simulations of the Maxwell system in a nonlinear media is presented.

Cited in 8 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L70 Second-order nonlinear hyperbolic equations
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78A25 Electromagnetic theory (general)
35Q60 PDEs in connection with optics and electromagnetic theory

Keywords:

Finite differences; stochastic equations; nonlocality; electromagnetic shocks; numerical examples; nonlinear wave equations; Maxwell equations; nonlinear media
Cite Review PDF
Full Text: DOI

References:

[1] DOI: 10.1137/0136033 · Zbl 0408.65075 · doi:10.1137/0136033
[2] Agrawal G. P., Nonlinear Fiber Optics (1995) · Zbl 1024.78514
[3] DOI: 10.1063/1.165948 · Zbl 1055.35505 · doi:10.1063/1.165948
[4] Alfimov G. L., Phys. Rev. 52 pp 4503– · doi:10.1103/PhysRevB.52.4503
[5] G. L. Alfimov, Nonlinear Klein–Gordon and Schrödinger Systems: Theory and Applications, eds. L. Vázquez, L. Streit and V. M. Pérez-García (World Scientific, Singapore, 1996) pp. 257–261.
[6] DOI: 10.1088/0951-7715/13/5/313 · Zbl 0969.35014 · doi:10.1088/0951-7715/13/5/313
[7] Bachiri H., Appl. Math. Comput.
[8] DOI: 10.1016/0370-1573(88)90015-4 · doi:10.1016/0370-1573(88)90015-4
[9] DOI: 10.1007/BF02188219 · Zbl 0842.58020 · doi:10.1007/BF02188219
[10] Berestycki H., Arch. Rat. Mech. Anal. 82 pp 313–
[11] DOI: 10.1137/S0036139996313447 · Zbl 0940.35035 · doi:10.1137/S0036139996313447
[12] Bishop A. R., Phys. Rev. 40 pp 6463– · doi:10.1103/PhysRevA.40.6463
[13] DOI: 10.1142/2647 · doi:10.1142/2647
[14] Blanchard Ph., Annales de l’Institut Henri Poincaré 47 pp 309–
[15] DOI: 10.1109/3.40655 · doi:10.1109/3.40655
[16] DOI: 10.1103/PhysRevLett.56.1952 · doi:10.1103/PhysRevLett.56.1952
[17] DOI: 10.1103/PhysRevLett.78.3282 · doi:10.1103/PhysRevLett.78.3282
[18] DOI: 10.1007/BF01217349 · Zbl 0579.35025 · doi:10.1007/BF01217349
[19] Bridges Th. J., Phys. Lett. 284 pp 184– · Zbl 0984.37104 · doi:10.1016/S0375-9601(01)00294-8
[20] Buchen W., J. Mech. 14 pp 597–
[21] Butcher J. C., Numerical Analysis of Ordinary Differential Equations. Runge–Kutta and General Methods (1987) · Zbl 0616.65072
[22] Buttiker M., Phys. Rev. 28 pp 1268– · doi:10.1103/PhysRevB.28.1268
[23] DOI: 10.1103/PhysRevLett.78.223 · doi:10.1103/PhysRevLett.78.223
[24] Caputo J., Phys. Rev. 44 pp 6219– · doi:10.1103/PhysRevA.44.6219
[25] DOI: 10.1007/BF01212340 · Zbl 0596.35117 · doi:10.1007/BF01212340
[26] Cazenave T., Textos Matemáticos 22, in: An Introduction to Nonlinear Schrödinger Equations (1989)
[27] DOI: 10.1088/0951-7715/3/2/001 · Zbl 0704.65052 · doi:10.1088/0951-7715/3/2/001
[28] Chubykalo O. A., Phys. Rev. 47 pp 7971– · doi:10.1103/PhysRevB.47.7971
[29] Cunha M. D., Phys. Lett. 221 pp 317– · Zbl 0972.35520 · doi:10.1016/0375-9601(96)00595-6
[30] Dahlqvist P., Phys. Rev. Lett. 23 pp 2837–
[31] DOI: 10.1016/0021-9991(81)90052-8 · Zbl 0477.65086 · doi:10.1016/0021-9991(81)90052-8
[32] Dodd R. K., Solitons and Nonlinear Wave Equations (1982) · Zbl 0496.35001
[33] DOI: 10.1007/BF02083817 · Zbl 0846.35001 · doi:10.1007/BF02083817
[34] DOI: 10.1016/0096-3003(91)90087-4 · Zbl 0732.65107 · doi:10.1016/0096-3003(91)90087-4
[35] Fei Z., Comp. Appl. Math. 10 pp 59–
[36] Fei Z., Phys. Lett. 159 pp 318–
[37] DOI: 10.1103/PhysRevA.46.5214 · doi:10.1103/PhysRevA.46.5214
[38] DOI: 10.1016/0096-3003(94)00152-T · Zbl 0832.65136 · doi:10.1016/0096-3003(94)00152-T
[39] Feng K., J. Comput. Math. 4 pp 279–
[40] Feng K., J. Comput. Math. 11 pp 71–
[41] DOI: 10.1103/PhysRevLett.76.2488 · doi:10.1103/PhysRevLett.76.2488
[42] Frenkel J., Fiz. Zhurnal 1 pp 137–
[43] DOI: 10.1016/S0377-0427(00)00527-6 · Zbl 0989.65099 · doi:10.1016/S0377-0427(00)00527-6
[44] García J. A., Revista Telefónica 19 pp 73–
[45] Gilles L., Phys. Rev. 60 pp 1051–
[46] DOI: 10.1006/jcph.2000.6460 · Zbl 0960.78015 · doi:10.1006/jcph.2000.6460
[47] DOI: 10.1016/0021-9991(84)90081-0 · Zbl 0561.65056 · doi:10.1016/0021-9991(84)90081-0
[48] Grønbech-Jensen N., Phys. Rev. 43 pp 5698– · doi:10.1103/PhysRevB.43.5698
[49] DOI: 10.1016/0370-1573(85)90073-0 · doi:10.1016/0370-1573(85)90073-0
[50] Guo B.-Y., App. Math. Comput. 14 pp 313–
[51] Guo B.-Y., App. Math. Comput. 18 pp 1–
[52] DOI: 10.1006/jcph.1996.0242 · Zbl 0866.65086 · doi:10.1006/jcph.1996.0242
[53] Hairer E., Solving Ordinary Differential Equations (1993) · Zbl 0789.65048
[54] Haraux A., Lecture Notes in Mathematics 841, in: Nonlinear Evolution Equations. Global Behaviour of Solutions (1981) · doi:10.1007/BFb0089606
[55] Ivanchenko Yu. M., Phys. Lett. 147 pp 65– · doi:10.1016/0375-9601(90)90015-G
[56] DOI: 10.1364/OL.15.001428 · doi:10.1364/OL.15.001428
[57] Jiménez S., Appl. Math. Comput. 25 pp 207–
[58] DOI: 10.1016/0096-3003(90)90091-G · Zbl 0697.65090 · doi:10.1016/0096-3003(90)90091-G
[59] S. Jiménez and L. Vázquez, Nonlinear Problems in Future Particle Accelerators, eds. W. Scandale and G. Turchetti (World Scientific, Singapore, 1991) pp. 151–162.
[60] DOI: 10.1016/0096-3003(94)90137-6 · Zbl 0806.65081 · doi:10.1016/0096-3003(94)90137-6
[61] DOI: 10.1016/S0096-3003(97)10126-6 · Zbl 0943.65087 · doi:10.1016/S0096-3003(97)10126-6
[62] Jiménez S., Appl. Math. Comput.
[63] DOI: 10.1103/RevModPhys.61.763 · doi:10.1103/RevModPhys.61.763
[64] DOI: 10.1364/OL.18.000485 · doi:10.1364/OL.18.000485
[65] Konotop V. V., Phys. Rev. 44 pp 2554– · doi:10.1103/PhysRevB.44.2554
[66] DOI: 10.1142/2320 · doi:10.1142/2320
[67] Konotop V. V., Phys. Lett. 236 pp 314– · doi:10.1016/S0375-9601(97)00682-8
[68] Legrand O., Phys. Rev. 36 pp 5068– · doi:10.1103/PhysRevA.36.5068
[69] Legrand O., Phys. Rev. 35 pp 3522– · doi:10.1103/PhysRevA.35.3522
[70] Lomdahl P., Phys. Rev. 34 pp 664– · doi:10.1103/PhysRevA.34.664
[71] Lomdahl P., Phys. Lett. 128 pp 427– · doi:10.1016/0375-9601(88)90123-5
[72] Luneburg R. K., Mathematical Theory of Optics (1964) · Zbl 0038.40103
[73] R. Mannela, Supercomputation in Nonlinear and Disordered Systems, eds. L. Vázquez, F. Tirado and I. Martín (World Scientific, Singapore, 1996) pp. 100–129.
[74] Martín I., Parall. Comput. 22 pp 1169–
[75] Martín I., SIAM News 12 pp 12–
[76] Matsuo T., RIMS Preprint No. 1280 (2000)
[77] DOI: 10.1007/BF01385708 · Zbl 0831.65099 · doi:10.1007/BF01385708
[78] McLaughlin D. W., Phys. Rev. 18 pp 1652– · doi:10.1103/PhysRevA.18.1652
[79] DOI: 10.1016/0030-4018(91)90415-A · doi:10.1016/0030-4018(91)90415-A
[80] DOI: 10.1103/PhysRevB.51.3054 · doi:10.1103/PhysRevB.51.3054
[81] Naumkin P. I., Nonlinear Nonlocal Equations in the Theory of Waves 133 (1994) · Zbl 0802.35002
[82] DOI: 10.1103/PhysRevLett.49.1883 · doi:10.1103/PhysRevLett.49.1883
[83] Parrondo J. M., J. Phys. 23 pp 2363–
[84] Pascual P. J., Comportamiento de un solitón de Sine–Gordon en presencia de perturbaciones estocásticas pequeñas (1984)
[85] Pascual P. J., Phys. Rev. 32 pp 8305– · doi:10.1103/PhysRevB.32.8305
[86] Pascual P. J., Hadronic J. 9 pp 307–
[87] DOI: 10.1007/3-540-59178-8_33 · doi:10.1007/3-540-59178-8_33
[88] DOI: 10.1016/0029-5582(62)90774-5 · Zbl 0106.20105 · doi:10.1016/0029-5582(62)90774-5
[89] Peyrard M., Physica 14 pp 88–
[90] DOI: 10.1103/PhysRevLett.85.2104 · doi:10.1103/PhysRevLett.85.2104
[91] DOI: 10.1364/OL.26.000376 · doi:10.1364/OL.26.000376
[92] Potter D., Computational Physics (1977) · Zbl 0314.70008
[93] W. H. Press, Numerical Recipes in C, 2nd edn. (Cambridge University Press, Cambridge, 1995) p. 841.
[94] DOI: 10.1007/978-3-642-96807-5 · doi:10.1007/978-3-642-96807-5
[95] Ritchmyer R., Difference Methods for Initial-Value Problems (1967)
[96] Rodríguez M. J., Phys. Rev. 41 pp 11437– · doi:10.1103/PhysRevB.41.11437
[97] DOI: 10.1103/PhysRev.139.A539 · doi:10.1103/PhysRev.139.A539
[98] Ruth R. D., IEEE Trans. Nucl. Sci. 30 pp 2669– · doi:10.1109/TNS.1983.4332919
[99] A. Sánchez and L. Vázquez, Nonlinear Coherent Structures, Lecture Notes in Physics 353, eds. M. Barthes and J. Léon (Springer-Verlag, Berlin, 1989) pp. 251–260. · doi:10.1007/BFb0033641
[100] Sánchez A., Int. J. Mod. Phys. 5 pp 2825– · doi:10.1142/S0217979291001115
[101] Sánchez A., Phys. Rev. 44 pp 1086– · doi:10.1103/PhysRevA.44.1086
[102] DOI: 10.1093/imanum/6.1.25 · Zbl 0593.65087 · doi:10.1093/imanum/6.1.25
[103] DOI: 10.1142/S0218127401002250 · Zbl 1090.39505 · doi:10.1142/S0218127401002250
[104] DOI: 10.1063/1.528578 · Zbl 0692.45004 · doi:10.1063/1.528578
[105] DOI: 10.1063/1.533231 · Zbl 0986.34055 · doi:10.1063/1.533231
[106] Sherwin M., Phys. Rev. 32 pp 5536– · doi:10.1103/PhysRevB.32.5536
[107] Sherwin M., Physica 23 pp 62–
[108] DOI: 10.1103/PhysRevLett.61.970 · doi:10.1103/PhysRevLett.61.970
[109] DOI: 10.1016/0045-7825(92)90115-Z · Zbl 0764.73096 · doi:10.1016/0045-7825(92)90115-Z
[110] Sohos G., Il Nuovo Cimento 104 pp 339– · doi:10.1007/BF02728404
[111] DOI: 10.1007/978-1-4757-2272-7 · doi:10.1007/978-1-4757-2272-7
[112] DOI: 10.1016/0021-9991(78)90038-4 · Zbl 0387.65076 · doi:10.1016/0021-9991(78)90038-4
[113] J. Stubbe and L. Vázquez, Mathematics + Physics, Lectures on Recent Result 3, ed. L. Streit (World Scientific, Singapore, 1988) pp. 81–123.
[114] Taflove A., Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (1998) · Zbl 0903.65098
[115] DOI: 10.1016/S0096-3003(96)00019-7 · Zbl 0870.65115 · doi:10.1016/S0096-3003(96)00019-7
[116] DOI: 10.1016/0898-1221(96)00136-8 · Zbl 0858.65124 · doi:10.1016/0898-1221(96)00136-8
[117] DOI: 10.1007/BF01007642 · Zbl 0509.60057 · doi:10.1007/BF01007642
[118] Vázquez L., Il Nuovo Cimento 69 pp 63– · doi:10.1007/BF02721241
[119] Vázquez L., Anales de Física 83 pp 254–
[120] Vázquez L., Phys. Lett. 189 pp 454– · doi:10.1016/0375-9601(94)91209-2
[121] Vázquez L., Supercomputation in Nonlinear and Disordered Systems: Algorithms, Applications and Architectures (1997)
[122] L. Vázquez, Proc. IASTED Int. Conf.: Computer Systems and Applications, ed. A. Nsour (Acta Press, Anaheim, 1998) pp. 249–252.
[123] Vázquez L., Il Nuovo Cimento 112 pp 455– · doi:10.1007/BF03035857
[124] DOI: 10.1142/9789812792167_0031 · doi:10.1142/9789812792167_0031
[125] Vázquez L., J. Comput. Math. 19 pp 9–
[126] Vázquez L., J. Comput. Math. 19 pp 445–
[127] Vázquez L., J. Comput. Math.
[128] Vázquez L., Appl. Math. Comput.
[129] Wiesenfeld K., Phys. Rev. 36 pp 2483– · doi:10.1103/PhysRevB.36.2483
[130] DOI: 10.1063/1.335465 · doi:10.1063/1.335465
[131] Zhong G., Phys. Lett. 133 pp 134– · Zbl 1369.70038 · doi:10.1016/0375-9601(88)90773-6
[132] Zigmund A., Trigonometric Series 1 (1959)
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.