Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation. (English) Zbl 1128.37043
Summary: The aim of this paper is the accurate numerical study of the Kadomtsev-Petviashvili (KP) equation. In particular, we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end, we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step, we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
Kadomtsev-Petviashvili equation; nonlinear dispersive models; multiple scales expansion; modulation theory; Davey-Stewartson systemSoftware:
MatlabReferences:
| [1] | M. J. Ablowitz and P. A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge, 1991. · Zbl 0762.35001 |
| [2] | M. J. Abramowiz and I. A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1972. · Zbl 0543.33001 |
| [3] | J. C. Alexander, R. L. Pego, and R. L. Sachs. On the transverse instability of solitary waves in the Kadomtsev–Petviashvili equation. Phys. Lett. A, 226:187–192, 1997. · Zbl 0962.35505 · doi:10.1016/S0375-9601(96)00921-8 |
| [4] | C. Besse, N. Mauser, and H. P. Stimming. Numerical study of the Davey–Stewartson system. Math. Model. Numer. Anal., 38(6):1035–1054, 2004. · Zbl 1080.65095 · doi:10.1051/m2an:2004049 |
| [5] | H. S. Bhat and R. C. Fetecau. A Hamiltonian regularization of the Burgers equation. J. Nonlinear Sci., 16:615–638, 2006. · Zbl 1108.35107 · doi:10.1007/s00332-005-0712-7 |
| [6] | V. N. Bogaevskiĭ. On Korteweg–de Vries, Kadomtsev–Petviashvili, and Boussinesq equations in the theory of modulations. Zh. Vychisl. Mat. Mat. Fiz., 30(10):1487–1501, 1990; translation in U.S.S.R. Comput. Math. Math. Phys., 30(5):148–159, 1991. |
| [7] | M. Boiti, F. Pempinelli, and A. Pogrebkov. Properties of solutions of the Kadomtsev–Petviashvili I equation. J. Math. Phys., 35(9):4683–4718, 1994. · Zbl 0814.35116 · doi:10.1063/1.530808 |
| [8] | A. de Bouard and Y. Martel. Nonexistence of L 2-compact solutions of the Kadomtsev–Petviashvili II equation. Math. Ann., 328:525–544, 2004. · Zbl 1330.35389 · doi:10.1007/s00208-003-0498-6 |
| [9] | J. Bourgain. On the Cauchy problem for the Kadomtsev–Petviashvili equation. Geom. Funct. Anal., 3(4):315–341, 1993. · Zbl 0787.35086 · doi:10.1007/BF01896259 |
| [10] | R. Camassa and D. D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71:1661–1665, 1993. · doi:10.1103/PhysRevLett.71.1661 |
| [11] | C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods in Fluid Dynamics. Springer, Berlin, 1988. · Zbl 0658.76001 |
| [12] | J. Colliander, C. Kenig, and G. Staffilani. Low regularity solutions for the Kadomtsev–Petviashvili I equation. Geom. Funct. Anal., 13(4):737–794, 2003. · Zbl 1039.35097 · doi:10.1007/s00039-003-0429-4 |
| [13] | T. Driscoll. A composite Runge–Kutta method for the spectral solution of semilinear PDEs. J. Comput. Phys., 182(2):357–367, 2002. · Zbl 1015.65050 · doi:10.1006/jcph.2002.7127 |
| [14] | B. F. Feng, T. Kawahara, and T. A. Mitsui. Conservative spectral method for several two-dimensional nonlinear wave equations. J. Comput. Phys., 153(2):467–487, 1999. · Zbl 0937.65108 · doi:10.1006/jcph.1999.6286 |
| [15] | E. V. Ferapontov and K. R. Khusnutdinova. Double waves in multi-dimensional systems of hydrodynamic type: The necessary condition for integrability. Preprint, 2004, arXiv: nlin.SI/0412064. · Zbl 1071.35118 |
| [16] | A. S. Fokas and L.-Y. Sung. On the solvability of the N-wave, Davey–Stewartson and Kadomtsev–Petviashvili equations. Inverse Probl., 8:673–708, 1992. · Zbl 0768.35069 · doi:10.1088/0266-5611/8/5/002 |
| [17] | A. S. Fokas and L.-Y. Sung. The Cauchy problem for the Kadomtsev–Petviashvili-I equation without the zero mass constraint. Math. Proc. Camb. Phil. Soc., 125:113–138, 1999. · Zbl 0923.35152 · doi:10.1017/S0305004198002850 |
| [18] | T. Grava and F.-R. Tian. The generation, propagation, and extinction of multiphases in the KdV zero-dispersion limit. Comm. Pure Appl. Math., 55(12):1569–1639, 2002. · Zbl 1024.37045 · doi:10.1002/cpa.10050 |
| [19] | T. Grava and C. Klein. Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations. Preprint, 2005, arXiv: math-ph/0511011. · Zbl 1139.65069 |
| [20] | R. H. Hardin and F. D. Tappert. Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations. SIAM Rev. Chron., 15:423, 1973. |
| [21] | G. Huang, V. A. Makarov, and M. G. Velarde. Two-dimensional solitons in Bose–Einstein condensates with a disk-shaped trap. Phys. Rev. A, 67:23604–23616, 2003. · doi:10.1103/PhysRevA.67.023604 |
| [22] | E. Infeld, A. Senatorski, and A. A. Skorupski. Numerical simulations of Kadomtsev–Petviashvili soliton interactions. Phys. Rev. E, 51:3183–3191, 1995. · doi:10.1103/PhysRevE.51.3183 |
| [23] | E. Infeld, A. A. Skorupski, and G. Rowlands. Instabilities and oscillations of one- and two-dimensional Kadomtsev–Petviashvili waves and solitons. II: Linear to nonlinear analysis. Roy. Soc. Lond. Proc. Ser. A (Math. Phys. Eng. Sci.), 458(2021):1231–1244, 2002. · Zbl 0998.37023 · doi:10.1098/rspa.2001.0937 |
| [24] | R. S. Johnson. The classical problem of water waves: a reservoir of integrable and nearly integrable equations. J. Nonlinear Math. Phys., 10:72–92, 2003. · Zbl 1362.35264 · doi:10.2991/jnmp.2003.10.s1.6 |
| [25] | C. A. Jones and P. H. Roberts. Motions in a Bose condensate, IV: Axisymmetric solitary waves. J. Phys. A Math. Gen., 15:2599–2619, 1982. · doi:10.1088/0305-4470/15/8/036 |
| [26] | B. B. Kadomtsev and V. I. Petviashvili. On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl., 15:539–541, 1970. · Zbl 0217.25004 |
| [27] | A.-K. Kassam and L. N. Trefethen. Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput., 26:1214–1233. 2005. · Zbl 1077.65105 · doi:10.1137/S1064827502410633 |
| [28] | D. Kaya and E. M. Salah. Numerical soliton-like solutions of the potential Kadomtsev–Petviashvili equation by the decomposition method. Phys. Lett. A, 320(2–3):92–199, 2003. · Zbl 1065.35219 · doi:10.1016/j.physleta.2003.11.021 |
| [29] | C. Klein. Fourth-order time-stepping for low dispersion Korteweg–de Vries and nonlinear Schrödinger equation. Preprint, 2006, available at http://www.mis.mpg.de/preprints.html . |
| [30] | Y. Kodama. A method for solving the dispersionless KP hierarchy and its exact solutions. Phys. Lett. A, 129(4):223–226, 1988. · doi:10.1016/0375-9601(88)90354-4 |
| [31] | Y. Kodama and J. Gibbons. A method for solving the dispersionless KP hierarchy and its exact solutions. II. Phys. Lett. A, 135(3):167–170, 1989. · doi:10.1016/0375-9601(89)90255-7 |
| [32] | I. M. Krichever. The averaging method for two-dimensional integrable equations. Funkt. Anal. Priloz., 22(3):37–52, 1988; translation in Funct. Anal. Appl. 22(3):200–213, 1989. |
| [33] | J. D. Lawson. Generalized Runge–Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal., 4:372–380, 1967. · Zbl 0223.65030 · doi:10.1137/0704033 |
| [34] | P. D. Lax and C. D. Levermore. The small dispersion limit of the Korteweg–de Vries equation I, II, III. Comm. Pure Appl. Math., 36:253–290, 571–593, 809–830, 1983. · Zbl 0532.35067 · doi:10.1002/cpa.3160360302 |
| [35] | C. D. Levermore. The hyperbolic nature of the zero dispersion KdV limit. Comm. Part. Differ. Equ., 13(4):495–514, 1988. · Zbl 0678.35081 · doi:10.1080/03605308808820550 |
| [36] | D. W. McLaughlin and J. A. Strain. Computing the weak limit of KdV. Comm. Pure Appl. Math., 47(10):1319–1364, 1994. · Zbl 0837.65111 · doi:10.1002/cpa.3160471003 |
| [37] | B. Minchev and W. Wright. A review of exponential integrators for first-order semi-linear problems. Technical Report 2, Norwegian University of Science and Technology, 2005. |
| [38] | L. Molinet, J. C. Saut, and N. Tzvetkov. Well-posedness and ill-posedness results for the Kadomtsev–Petviashvili-I equation. Duke Math. J., 115(2):353–384, 2002. · Zbl 1033.35103 · doi:10.1215/S0012-7094-02-11525-7 |
| [39] | L. Molinet, J. C. Saut, and N. Tzvetkov. Remarks on the mass constraint for KP type equations. Preprint, 2006, arXiv: math/0603303. · Zbl 1139.35009 |
| [40] | S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov. Theory of Solitons: The Inverse Scattering Method. Springer Monographs in Contemporary Mathematics. Springer, New York, 1984. · Zbl 0598.35002 |
| [41] | A. Rozanova. Mathematical analysis of the Khokhlov–Zabolotskaya–Kuznetsov equation. Preprint, 2006. · Zbl 1118.35036 |
| [42] | G. Schneider. Approximation of the Korteweg–de Vries equation by the nonlinear Schrödinger equation. J. Differ. Equ., 147(2):333–354, 1998. · Zbl 0940.35179 · doi:10.1006/jdeq.1998.3417 |
| [43] | A. Senatorski and E. Infeld. Simulations of two-dimensional Kadomtsev–Petviashvili soliton dynamics in three-dimensional space. Phys. Rev. Lett., 77:2855–2858, 1996. · doi:10.1103/PhysRevLett.77.2855 |
| [44] | C. Sulem and P. L. Sulem. The nonlinear Schrödinger equation. In Applied Mathematical Sciences, page 139. Springer, New York, 1999. · Zbl 0928.35157 |
| [45] | H. Takaoka. Global well-posedness for the Kadomtsev–Petviashvili II equation. Discr. Cont. Dyn. Syst., 6:483–499, 2000. · Zbl 1021.35099 · doi:10.3934/dcds.2000.6.483 |
| [46] | H. Takaoka and N. Tzvetkov. On the local regularity of the Kadomtsev–Petviashvili-II equation. Int. Math. Res. Notes, 2001(2):77–114, 2001. · Zbl 0977.35126 · doi:10.1155/S1073792801000058 |
| [47] | F. R. Tian. Oscillations of the zero dispersion limit of the Korteweg–de Vries equations. Comm. Pure Appl. Math., 46:1093–1129, 1993. · Zbl 0810.35114 · doi:10.1002/cpa.3160460802 |
| [48] | L. N. Trefethen. Spectral Methods in MATLAB. SIAM, Philadelphia, 2000. · Zbl 0953.68643 |
| [49] | http://www.comlab.ox.ac.uk/oucl/work/nick.trefethen . |
| [50] | S. Turitsyn and G. Falkovitch. Stability of magneto-elastic solitons and self-focusing of sound in antiferromagnets. Sov. Phys. J. Exp. Theor. Phys., 62:146–152, 1985. |
| [51] | N. Tzvetkov. Bilinear Estimates Related to the KP Equations, volume XIX of Journées Équations aux Dérivées Part. University Nantes, Nantes, 2000. |
| [52] | S. Venakides. The zero dispersion limit of the Koreteg–de Vries equation for initial data with nontrivial reflection coefficient. Comm. Pure Appl. Math.,38:125–155, 1985. · Zbl 0571.35095 · doi:10.1002/cpa.3160380202 |
| [53] | X. P. Wang, M. Ablowitz, and H. Segur. Wave collapse and instability of solitary waves of a generalized Kadomtsev–Petviashvili equation. Phys. D,78:241–265, 1994. · Zbl 0824.35116 · doi:10.1016/0167-2789(94)90118-X |
| [54] | A. Wazwaz. A computational approach to soliton solutions of the Kadomtsev–Petviashvili equation. Appl. Math. Comput., 123(2):205–217, 2001. · Zbl 1024.65098 · doi:10.1016/S0096-3003(00)00065-5 |
| [55] | V. E. Zakharov and A. E. Kuznetsov. Multi-scale expansion in the theory of systems integrable by the inverse scattering transform. Phys. D, 18:455–463, 1986. · Zbl 0611.35079 · doi:10.1016/0167-2789(86)90214-9 |
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.
