×

On critical behaviour in systems of Hamiltonian partial differential equations. (English) Zbl 1321.35208

Authors’ abstract: We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (\(P_I\)) equation or its fourth-order analogue \(P_I^2\). As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies

Software:

fminsearch; bvp4c; Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agrawal, G.P.: Nonlinear Fiber Optics, 4th edn. Academic Press, San Diego (2006) · Zbl 1024.78514
[2] Alinhac, S.: Blowup for Nonlinear Hyperbolic Equations. Progress in Nonlinear Differential Equations and their Applications, 17. Birkhäuser Boston Inc, Boston (1995) · Zbl 0820.35001
[3] Arnold, V.I., Goryunov, V.V., Lyashko, O.V., Vasil’ev, V.A.: Singularity Theory. I. Dynamical systems. VI, Encyclopaedia Math. Sci. 6. Springer, Berlin (1993)
[4] Arsie, A., Lorenzoni, P., Moro, A.: Integrable viscous conservation laws, Preprint: http://xxx.lanl.gov/pdf/1301.0950 · Zbl 1319.37040
[5] Bambusi, D., Ponno, A.: Resonance, Metastability and Blow up in FPU. The Fermi-Pasta-Ulam Problem, pp. 191-205, Lecture Notes in Phys., 728, Springer, Berlin (2008) · Zbl 1151.82303
[6] Bao, W., Jin, S., Markowich, P.A.: On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175, 487-524 (2002) · Zbl 1006.65112 · doi:10.1006/jcph.2001.6956
[7] Bao, W., Jin, S., Markowich, P.A.: Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semi-classical regimes, SIAM J. Sci. Comput. pp. 27-64 (2003) · Zbl 1038.65099
[8] Benettin, G., Ponno, A.: Time-scales to equipartition in the Fermi-Pasta-Ulam problem: finite-size effects and thermodynamic limit. J. Stat. Phys. 144(4), 793-812 (2011) · Zbl 1227.82008 · doi:10.1007/s10955-011-0277-9
[9] Berland, H., Skaflestad, B.: Solving the nonlinear Schrödinger equation using exponential integrators, Technical Report 3/05, The Norwegian Institute of Science and Technology (2005). http://www.math.ntnu.no/preprint/ · Zbl 1109.65060
[10] Berland, H., Islas, A.L., Schober, C.M.: Solving the nonlinear Schrödinger equation using exponential integrators. J. Comput. Phys. 255, 284-299 (2007) · Zbl 1122.65125 · doi:10.1016/j.jcp.2006.11.030
[11] Berry, M.V., Nye, J.F., Wright, F.J.: The elliptic umbilic diffraction catastrophe. Philos. Trans. R. Soc. Lond. Ser. A 291, 453-484 (1979) · doi:10.1098/rsta.1979.0039
[12] Bertola, M., Tovbis, A.: Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painlevé equation. Preprint http://xxx.lanl.gov/pdf/1108.0321 · Zbl 1320.33029
[13] Bertola, M., Tovbis, A.: Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the Tritronquée solution to Painlevé-I. Commun. Pure Appl. Math. 66(5), 678-752 (2013) · Zbl 1355.35169 · doi:10.1002/cpa.21445
[14] Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150(1), 185-266 (1999) · Zbl 0956.42014 · doi:10.2307/121101
[15] Bourgain, J.: Global Solutions of Nonlinear Schrödinger Equations. American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI (1999). viii+182 pp. ISBN: 0-8218-1919-4 · Zbl 1108.53044
[16] Boutroux, P.: Recherches sur les transcendants de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre. Ann. École Norm 30, 265-375 (1913) · JFM 44.0382.02
[17] Bressan, A.: Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem. Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford (2000) · Zbl 0997.35002
[18] Brézin, É., Marinari, E., Parisi, G.: A nonperturbative ambiguity free solution of a string model. Phys. Lett. B 242, 35-38 (1990) · doi:10.1016/0370-2693(90)91590-8
[19] Bronski, J.C., Kutz, J.N.: Numerical simulation of the semiclassical limit of the focusing nonlinear Schrödinger equation. Phys. Lett., A 254, 325-336 (2002) · doi:10.1016/S0375-9601(99)00133-4
[20] Buckingham, R.J., Miller, P.D.: The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix. J. Anal. Math. 118(2), 397-492 (2012) · Zbl 1307.35255 · doi:10.1007/s11854-012-0041-3
[21] Buckingham, R., Venakides, S.: Long-time asymptotics of the nonlinear Schrödinger equation shock problem. Comm. Pure Appl. Math., 60(9), 1349-1414 (2007) · Zbl 1125.35089
[22] Carles, R.: On the semi-classical limit for the nonlinear Schrödinger equation. In: Stationary and Time Dependent Gross-Pitaevskii Equations. Contemporary Mathematics, vol. 473, pp. 105-127. American Mathematical Society, Providence, RI (2008) · Zbl 1166.35372
[23] Ceniceros, H.D.: A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Commun. Pure Appl. Anal. 1, 1-18 (2002) · Zbl 1010.35098 · doi:10.3934/cpaa.2002.1.1
[24] Ceniceros, H.D., Tian, F.-R.: A numerical study of the semi-classical limit of the focusing nonlinear Schrödinger equation. Phys. Lett. A 306, 25-34 (2002) · Zbl 1005.81029 · doi:10.1016/S0375-9601(01)00011-1
[25] Claeys, T., Grava, T.: Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach. Commun. Math. Phys. 286(3), 979-1009 (2009) · Zbl 1173.35654 · doi:10.1007/s00220-008-0680-5
[26] Claeys, T., Vanlessen, M.: Universality of a double scaling limit near singular edge points in random matrix models. Commun. Math. Phys. 273(2), 499-532 (2007) · Zbl 1136.82023 · doi:10.1007/s00220-007-0256-9
[27] Claeys, T., Vanlessen, M.: The existence of a real pole-free solution of the fourth order analogue of the Painlevé-I equation. Nonlinearity 20(5), 1163-1184 (2007) · Zbl 1175.33016 · doi:10.1088/0951-7715/20/5/006
[28] Conti, C., Fratalocchi, A., Peccianti, M., Ruocco, G., Trillo, S.: Observation of a gradient catastrophe generating solitons. Phys. Rev. Lett. 102, 083902 (2009) · doi:10.1103/PhysRevLett.102.083902
[29] Costin, O.: Correlation between pole location and asymptotic behavior for Painlevé-I solutions. Commun. Pure Appl. Math. 52, 461-478 (1999) · Zbl 0910.34003 · doi:10.1002/(SICI)1097-0312(199904)52:4<461::AID-CPA3>3.0.CO;2-T
[30] Costin, O., Huang, M., Tanveer, S.: Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of PI. Duke Math. J. 163(4), 665-704 (2014) · Zbl 1305.34151
[31] Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851-1112 (1993) · Zbl 1371.37001 · doi:10.1103/RevModPhys.65.851
[32] de Bouard, A.: Analytic solutions to nonelliptic nonlinear Schrödinger equations. J. Differ. Equ. 104(1), 196-213 (1993) · Zbl 0798.35138 · doi:10.1006/jdeq.1993.1069
[33] Degiovanni, L., Magri, F., Sciacca, F.V.: On deformation of poisson manifolds of hydrodynamic type. Commun. Math. Phys. 253(1), 1-24 (2005) · Zbl 1108.53044 · doi:10.1007/s00220-004-1190-8
[34] Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes 3. New York University (1999)
[35] Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335-1425 (1999) · Zbl 0944.42013 · doi:10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
[36] Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491-1552 (1999) · Zbl 1026.42024 · doi:10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.0.CO;2-#
[37] Deift, P., McLaughlin, K.T.-R.: A continuum limit of the Toda lattice. Mem. Am. Math. Soc. 131(624), x+216 pp (1998) · Zbl 0946.37035
[38] Deift, P., Venakides, S., Zhou, X.: New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Int. Math. Res. Notices 6, 286-299 (1997) · Zbl 0873.65111 · doi:10.1155/S1073792897000214
[39] Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. (2) 137(2), 295-368 (1993) · Zbl 0771.35042 · doi:10.2307/2946540
[40] Deift, P., Zhou, X.: Perturbation theory for infinite-dimensional integrable systems on the line. A case study. Acta Math. 188(2), 163-262 (2002) · Zbl 1006.35089 · doi:10.1007/BF02392683
[41] Degasperis, A.: Multiscale Expansion and Integrability of Dispersive Wave Equations. Integrability, Lecture Notes in Phys., 767, pp. 215-244. Springer, Berlin (2009) · Zbl 1167.35046
[42] DiFranco, J., Miller, P.D.: The semiclassical modified nonlinear Schrödinger equation. I. Modulation theory and spectral analysis. Phys. D 237(7), 947-997 (2008) · Zbl 1158.35086 · doi:10.1016/j.physd.2007.11.022
[43] Driscoll, T.: A composite Runge-Kutta method for the spectral solution of semilinear PDEs. J. Comput. Phys. 182, 357-367 (2002) · Zbl 1015.65050 · doi:10.1006/jcph.2002.7127
[44] Dubrovin, B.: On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour. Commun. Math. Phys. 267, 117-139 (2006) · Zbl 1109.35070 · doi:10.1007/s00220-006-0021-5
[45] Dubrovin, B.: On universality of critical behaviour in Hamiltonian PDEs. Geometry, topology, and mathematical physics, pp. 59-109, Am. Math. Soc. Transl. Ser. 2, 224, Am. Math. Soc., Providence, RI (2008) · Zbl 1172.35001
[46] Dubrovin, B., Elaeva, M.: On the critical behavior in nonlinear evolutionary PDEs with small viscosity. Russ. J. Math. Phys. 19(4), 449-460 (2012) · Zbl 1263.35019 · doi:10.1134/S106192081204005X
[47] Dubrovin, B., Grava, T., Klein, C.: On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation. J. Nonlinear Sci. 19(1), 57-94 (2009) · Zbl 1220.37048 · doi:10.1007/s00332-008-9025-y
[48] Dubrovin, B., Grava, T., Klein, C.: Numerical study of break-up in generalized Korteweg-de Vries and Kawahara equations. SIAM J. Appl. Math. 71, 983-1008 (2011) · Zbl 1231.65175 · doi:10.1137/100819783
[49] Dubrovin, B., Liu, S.-Q., Zhang, Y.: On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations. Commun. Pure Appl. Math. 59, 559-615 (2006) · Zbl 1108.35112 · doi:10.1002/cpa.20111
[50] Dubrovin, B., Novikov, S.: Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. Russ. Math. Surveys 44(6), 35-124 (1989) · Zbl 0712.58032 · doi:10.1070/RM1989v044n06ABEH002300
[51] Duits, M., Kuijlaars, A.: Painlevé-I asymptotics for orthogonal polynomials with respect to a varying quartic weight. Nonlinearity 19(10), 2211-2245 (2006) · Zbl 1129.34059 · doi:10.1088/0951-7715/19/10/001
[52] El, G.A.: Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos 15(3), 037103 (2005). 21 pp · Zbl 1144.37341
[53] Falqui, G.: On a Camassa-Holm type equation with two dependent variables. J. Phys. A 39(2), 327-342 (2006) · Zbl 1084.37053 · doi:10.1088/0305-4470/39/2/004
[54] Fokas, A.S., Its, A.R., Kitaev, A.V.: Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys. 142, 313-344 (1991) · Zbl 0742.35047 · doi:10.1007/BF02102066
[55] Forest, M.G., Lee, J.E.: Geometry and modulation theory for the periodic nonlinear Schrödinger equation. In: Oscillation Theory, Computation, and Methods of Compensated Compactness (Minneapolis, Minn., 1985), pp. 35-69. The IMA Volumes in Mathematics and Its Applications, 2. Springer, New York (1986) · Zbl 0910.35115
[56] Gérard, P.: Remarques sur l’analyse semi-classique de l’équation de Schrödinger non linéaire. Séminaire sur les équations aux Dérivées Partielles, 1992-1993, Exp. No. XIII, 13 pp., École Polytech., Palaiseau (1993)
[57] Getzler, E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111(3), 535-560 (2002) · Zbl 1100.32008 · doi:10.1215/S0012-7094-02-11136-3
[58] Ghofraniha, N., Conti, C., Ruocco, G., Trillo, S.: Shocks in nonlocal media. Phys. Rev. Lett. 99, 043903 (2007) · doi:10.1103/PhysRevLett.99.043903
[59] Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32, 1-32 (1979) · Zbl 0396.35028 · doi:10.1016/0022-1236(79)90076-4
[60] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Translated from the Russian. Sixth edition. Translation edited and with a preface by Jeffrey, A., Zwillinger, D., Academic Press Inc, San Diego, CA (2000) · Zbl 0981.65001
[61] Grava, T., Klein, C. A.: Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions. Phys. D 241(23-24), 2246-2264 (2012) · Zbl 1257.35165
[62] Grava, T., Klein, C.: Numerical study of a multiscale expansion of KdV and Camassa-Holm equation. In: Baik, J., Kriecherbauer, T., Li, L.-C., McLaughlin, K.D.T-R., Tomei. C. (eds.) Integrable Systems and Random Matrices. Contemp. Math. vol. 458, 81-99 (2008) · Zbl 1157.35471
[63] Grava, T., Klein, C.: Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations. Commun. Pure Appl. Math. 60(11), 1623-1664 (2007) · Zbl 1139.65069 · doi:10.1002/cpa.20183
[64] Grenier, E.: Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Am. Math. Soc. 126, 523-530 (1998) · Zbl 0910.35115 · doi:10.1090/S0002-9939-98-04164-1
[65] Grinevich, P., Novikov, S.P.: String equation. II. Physical solution. (Russian) Algebra i Analiz 6(3), 118-140 (1994); translation in St. Petersburg Math. J. 6(3), 553-574 (1995) · Zbl 0836.35142
[66] Gurevich, A.G., Pitaevskii, L.P.: Non stationary structure of a collisionless shock waves. JEPT Lett. 17, 193-195 (1973)
[67] Henrici, A., Kappeler, T.: Resonant normal form for even periodic FPU chains. J. Eur. Math. Soc. 11(5), 1025-1056 (2009) · Zbl 1181.37081 · doi:10.4171/JEMS/174
[68] Hoefer, M.A., Ilan, B.: Dark solitons, dispersive shock waves, and transverse instabilities. Multiscale Model. Simul. 10(2), 306-341 (2012) · Zbl 1248.35195 · doi:10.1137/110834822
[69] Hou, T.Y., Lax, P.D.: Dispersive approximations in fluid dynamics. Commun. Pure Appl. Math. 44, 1-40 (1991) · Zbl 0729.76065 · doi:10.1002/cpa.3160440102
[70] Il’in, A.M.: Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. AMS Translations of Mathematical Monographs, vol. 102, 281 pp (1992) · Zbl 0754.34002
[71] Ince, E.L.: Ordinary Differential Equations. Dover Publications, New York (1944) · Zbl 0063.02971
[72] Jenkins, R., Ken, D., McLaughlin, T.-R.: Semiclassical limit of focusing NLS for a family of square barrier initial data. Commun. Pure Appl. Math. 67(2), 246-320 (2014) · Zbl 1332.35327 · doi:10.1002/cpa.21494
[73] Jin, S., Levermore, C.D., McLaughlin, D.W.: The behavior of solutions of the NLS equation in the semiclassical limit. Singular Limits of Dispersive Waves (Lyon, 1991), 235-255, NATO Adv. Sci. Inst. Ser. B Phys., 320, Plenum, New York (1994) · Zbl 0849.35130
[74] Jin, S., Levermore, C.D., McLaughlin, D.W.: The semiclassical limit of the defocusing NLS hierarchy. Commun. Pure Appl. Math. 52, 613-654 (1999) · Zbl 0935.35148 · doi:10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L
[75] Joshi, N., Kitaev, A.: On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107, 253-291 (2001) · Zbl 1152.34395 · doi:10.1111/1467-9590.00187
[76] Kamvissis, S.: Long time behavior for the focusing nonlinear Schrödinger equation with real spectral singularities. Commun. Math. Phys. 180, 325-341 (1996) · Zbl 0872.35101 · doi:10.1007/BF02099716
[77] Kamvissis, S., McLaughlin, K.D.T.-R., Miller, P.D.: Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation. Annals of Mathematics Studies, 154. Princeton University Press, Princeton (2003) · Zbl 1057.35063
[78] Kapaev, A.A.: Weakly nonlinear solutions of the equation \[{\rm P}^2_1\] P12, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187 (1991), Differentsialnaya Geom. Gruppy Li i Mekh. 12, 88-109, 172-173, 175; translation in J. Math. Sci. 73(4), 468-481 (1995) · Zbl 0746.34008
[79] Kapaev, A.: Quasi-linear Stokes phenomenon for the Painlevé first equation. J. Phys. A Math. Gen. 37, 11149-11167 (2004) · Zbl 1080.34071 · doi:10.1088/0305-4470/37/46/005
[80] Kapaev, A., Klein, C., Grava, T.: On the tritronquée solutions of \[P_I^2\] I2. Constr. Approx. (to appear). arXiv:1306.6161 · Zbl 1326.34136
[81] Kassam, A.-K., Trefethen, L.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214-1233 (2005) · Zbl 1077.65105 · doi:10.1137/S1064827502410633
[82] Kenig, C.E., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166(3), 645-675 (2006) · Zbl 1115.35125 · doi:10.1007/s00222-006-0011-4
[83] Kitaev, A.: The isomonodromy technique and the elliptic asymptotics of the first Painlevé transcendent. Algebra i Analiz 5(3), pp. 179-211 (1993); translation in St. Petersburg Math. J. 5(3), 577-605 (1994) · Zbl 0801.34009
[84] Klainerman, S., Majda, A.: Formation of singularities for wave equations including the nonlinear vibrating string. Commun. Pure Appl. Math. 33, 241-263 (1980) · Zbl 0443.35040 · doi:10.1002/cpa.3160330304
[85] Klein, C.: Fourth-order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation. Electron. Trans. Numer. Anal. 39, 116-135 (2008) · Zbl 1186.65134
[86] Kodama, Y., Mikhailov, A.: Obstacles to asymptotic integrability, algebraic aspects of integrable systems, 173-204, Progr. Nonlinear Differential Equations Appl., 26, Birkhäuser, Boston, MA (1997) · Zbl 0867.35091
[87] Kong, D.: Formation and propagation of singularities for \[2\times 22\]×2 quasilinear hyperbolic systems. Trans. Am. Math. Soc. 354(8), 3155-3179 (2002) · Zbl 1003.35079 · doi:10.1090/S0002-9947-02-02982-3
[88] Krasny, R.: A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65-93 (1986) · Zbl 0601.76038 · doi:10.1017/S0022112086002732
[89] Kudashev, V., Suleimanov, B.: A soft mechanism for the generation of dissipationless shock waves. Phys. Lett. A 221, 204-208 (1996) · doi:10.1016/0375-9601(96)00570-1
[90] Kuksin, S.B.: Perturbation theory for quasiperiodic solutions of infinite-dimensional Hamiltonian systems, and its application to the Korteweg-de Vries equation. Matem. Sbornik, 136 (1988) [Russian]. English translation in Math. USSR Sbornik 64, 397-413 (1989) · Zbl 0678.58037
[91] Kuksin, S.B., Poeschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schroedinger equation. Ann. Math. 143(1), 149-179 (1996) · Zbl 0847.35130 · doi:10.2307/2118656
[92] Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim. 9, 112-147 (1988) · Zbl 1005.90056 · doi:10.1137/S1052623496303470
[93] Lax, P., Levermore, D.: The small dispersion limit of the Korteweg-de Vries equation. I, II, III. Commun. Pure Appl. Math. 36, 253-290, 571-593, 809-829 (1983) · Zbl 0532.35067
[94] Lax, P.D., Levermore, C.D., Venakides, S.: The generation and propagation of oscillations in dispersive initial value problems and their limiting behavior. In: Important Developments in Soliton Theory, pp. 205-241, Springer Ser. Nonlinear Dynam., Springer, Berlin (1993) · Zbl 0819.35122
[95] Lee, S.-Y., Teodorescu, R., Wiegmann, P.: Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painlevé-I transcendent. Phys. D 240, 1080-1091 (2011) · Zbl 1218.76021 · doi:10.1016/j.physd.2010.09.017
[96] Liu, S.-Q., Wu, C.-Z., Zhang, Y.: On properties of Hamiltonian structures for a class of evolutionary PDEs. Lett. Math. Phys. 84(1), 47-63 (2008) · Zbl 1153.37418 · doi:10.1007/s11005-008-0234-y
[97] Liu, S.-Q., Zhang, Y.: On quasitriviality and integrability of a class of scalar evolutionary PDEs. J. Geom. Phys. 57, 101-119 (2006) · Zbl 1111.37056 · doi:10.1016/j.geomphys.2006.02.005
[98] Lorenzoni, P., Paleari, S.: Metastability and dispersive shock waves in the Fermi-Pasta-Ulam system. Phys. D 221(2), 110-117 (2006) · Zbl 1101.82019 · doi:10.1016/j.physd.2006.07.017
[99] Linares, F., Ponce, G.: Introduction to nonlinear dispersive equations. Universitext. Springer, New York (2009). xii+256 pp. ISBN: 978-0-387-84898-3 · Zbl 1178.35004
[100] Lyng, G.D., Miller, P.D.: The \[NN\]-soliton of the focusing nonlinear Schrödinger equation for \[NN\] large. Commun. Pure Appl. Math. 60, 951-1026 (2007) · Zbl 1185.35259 · doi:10.1002/cpa.20162
[101] Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Sciences, 53. Springer, New York (1984). viii+159 pp. ISBN: 0-387-96037-635L65 · Zbl 0537.76001
[102] Manakov, S.V., Santini, P.M.: On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking. J. Phys. A 44(40), 405203 (2011). 15 pp · Zbl 1241.35181 · doi:10.1088/1751-8113/44/40/405203
[103] Martínez-Alonso, L., Medina, E.: Regularization of Hele-Shaw flows, multiscaling expansions and the Painlevé-I equation. Chaos Solitons Fract. 41(3), 1284-1293 (2009) · Zbl 1198.35232 · doi:10.1016/j.chaos.2008.05.020
[104] Masoero, D., Raimondo, A.: Semiclassical limit for generalized KdV equations before the gradient catastrophe. Lett. Math. Phys. 103(5), 559-583 (2013) · Zbl 1291.35300 · doi:10.1007/s11005-013-0605-x
[105] Miller, P.D., Xu, Z.: The Benjamin-Ono hierarchy with asymptotically reflectionless initial data in the zero-dispersion limit. Commun. Math. Sci. 10(1), 117-130 (2012) · Zbl 1291.35229 · doi:10.4310/CMS.2012.v10.n1.a6
[106] Menikoff, A.: The existence of unbounded solutions of the Korteweg-de Vries equation. Commun. Pure Appl. Math. 25, 407-432 (1972) · Zbl 0226.35079 · doi:10.1002/cpa.3160250404
[107] Merle, F., Raphael, P.: On universality of blow-up profile for \[L^2\] L2 critical nonlinear Schrödinger equation. Invent. Math. 156, 565-672 (2004) · Zbl 1067.35110 · doi:10.1007/s00222-003-0346-z
[108] Métivier, G.: Remarks on the well-posedness of the nonlinear Cauchy problem. Geometric analysis of PDE and several complex variables, 337-356, Contemp. Math., 368, Am. Math. Soc., Providence, RI (2005) · Zbl 1071.35074
[109] Miller, P.D., Kamvissis, S.: On the semiclassical limit of the focusing nonlinear Schrödinger equation. Phys. Lett. A 247, 75-86 (1998) · Zbl 0941.81029 · doi:10.1016/S0375-9601(98)00565-9
[110] Moore, G.: Geometry of the string equations. Commun. Math. Phys. 133, 261-304 (1990) · Zbl 0727.35134 · doi:10.1007/BF02097368
[111] Newell, A.C.: Solitons in Mathematics and Physics. CBMS-NSF Regional Conference Series in Applied Mathematics, 48. SIAM, Philadelphia (1985)
[112] Novikov, S.P., Manakov, S.V., Pitaevskiĭ, L.P., Zakharov, V.E.: Theory of Solitons. The Inverse Scattering Method. Translated from the Russian. Contemporary Soviet Mathematics. Consultants Bureau [Plenum], New York (1984) · Zbl 0598.35002
[113] Rasmussen, P.D., Bang, O., Krolikowski, W.: Theory of nonlocal soliton interaction in nematic liquid cristals. Phys. Rev. E 72, 066611 (2005) · doi:10.1103/PhysRevE.72.066611
[114] Satsuma, J., Yajima, N.: Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media. Suppl. Prog. Theor. Phys. 55, 284-306 (1974) · doi:10.1143/PTPS.55.284
[115] Serre, D.: Systèmes de lois de conservation I : hyperbolicité, entropies, ondes de choc; Systèmes de lois de conservation II: structures géométriques, oscillation et problèmes mixtes, Paris Diderot Editeur (1996) · Zbl 0930.35002
[116] Shabat, A.B.: One-dimensional perturbations of a differential operator, and the inverse scattering problem. In: Problems in Mechanics and Mathematical Physics, pp. 279-296. Nauka, Moscow (1976)
[117] Shampine, L.F., Reichelt, M.W., Kierzenka, J.: Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c, available at http://www.mathworks.com/bvp_tutorial
[118] Sikivie, P.: The caustic ring singularity. Phys. Rev. D 60, 063501 (1999) · doi:10.1103/PhysRevD.60.063501
[119] Slemrod, M.: Monotone increasing solutions of the Painlevé 1 equation \[y^{\prime \prime }=y^2+x\] y″=y2+x and their role in the stability of the plasma-sheath transition. Eur. J. Appl. Math. 13, 663-680 (2002) · Zbl 1027.34044 · doi:10.1017/S0956792502004977
[120] Strachan, I.A.B.: Deformations of the Monge/Riemann hierarchy and approximately integrable systems. J. Math. Phys. 44, 251-262 (2003) · Zbl 1061.37052 · doi:10.1063/1.1522134
[121] Sulem, C., Sulem, P.: The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Applied Mathematical Sciences, 139. Springer, New York (1999) · Zbl 0928.35157
[122] Tao, T.: Why are soliton stable? Bull. Am. Math. Soc. 46(1), 1-33 (2009) · Zbl 1155.35082 · doi:10.1090/S0273-0979-08-01228-7
[123] Tao, T.: Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2006) · Zbl 1106.35001
[124] Thom, R.: Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Addison-Wesley, Reading (1989) · Zbl 0698.92001
[125] Tian, F.R.: The initial value problem for the Whitham averaged system. Commun. Math. Phys. 166(1), 79-115 (1994) · Zbl 0812.35131 · doi:10.1007/BF02099302
[126] Tian, F.R., Ye, J.: On the Whitham equations for the semiclassical limit of the defocusing nonlinear Schrödinger equation. Commun. Pure Appl. Math. 52(6), 655-692 (1999) · Zbl 0935.35158 · doi:10.1002/(SICI)1097-0312(199906)52:6<655::AID-CPA1>3.0.CO;2-A
[127] Tovbis, A., Venakides, S., Zhou, X.: On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation. Commun. Pure Appl. Math. 57, 877-985 (2004) · Zbl 1060.35137 · doi:10.1002/cpa.20024
[128] Tovbis, A., Venakides, S., Zhou, X.: On the long-time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schödinger equation: pure radiation case. Commun. Pure Appl. Math. 59, 1379-1432 (2006) · Zbl 1115.35127 · doi:10.1002/cpa.20142
[129] Trefethen, L.: Spectral Methods in MATLAB, vol. 10 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000) · Zbl 0953.68643
[130] Tsarev, S.: The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method. Math. USSR Izv. 37, 397-419 (1991) · Zbl 0796.76014 · doi:10.1070/IM1991v037n02ABEH002069
[131] Tsutsumi, \[Y.: L^2\] L2-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkcial. Ekvac. 30, 115-125 (1987) · Zbl 0638.35021
[132] Venakides, S.: The Korteweg-de Vries equation with small dispersion: higher order Lax-Levermore theory. Commun. Pure Appl. Math. 43(3), 335-361 (1990) · Zbl 0705.35125 · doi:10.1002/cpa.3160430303
[133] Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974) · Zbl 0373.76001
[134] Whitney, H.: On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane. Ann. Math. (2) 62, 374-410 (1955) · Zbl 0068.37101 · doi:10.2307/1970070
[135] Zabusky, N., Kruskal, M.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 2403 (1965) · Zbl 1201.35174 · doi:10.1103/PhysRevLett.15.240
[136] Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34(1), 62-69 (1972); translated from Ž. Eksper. Teoret. Fiz. 1, 118-134 (1971) · Zbl 0226.35079
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.