Rational solutions to the KPI equation and multi rogue waves. (English) Zbl 1378.35266
Summary: We construct here rational solutions to the Kadomtsev-Petviashvili equation (KPI) as a quotient of two polynomials in \(x\), \(y\) and \(t\) depending on several real parameters. This method provides an infinite hierarchy of rational solutions written in terms of polynomials of degrees \(2N(N+1)\) in \(x\), \(y\) and \(t\) depending on \(2N-2\) real parameters for each positive integer \(N\). We give explicit expressions of the solutions in the simplest cases \(N=1\) and \(N=2\) and we study the patterns of their modulus in the \((x,y)\) plane for different values of time \(t\) and parameters.
References:
| [1] | Solli, D. R.; Ropers, C.; Koonath, P.; Jalali, B., Nature, 450, 1054-1057 (2007) |
| [2] | Bludov, Y. V.; Konotop, V. V.; Akhmediev, N., Phys. Rev. A, 80 (2009), 033610-1-5 |
| [3] | Stenflo, L.; Marklund, M., J. Plasma Phys., 76, 3-4, 293-295 (2010) |
| [4] | Yan, Z. Y., Commun. Theor. Phys., 54, 5 (2010), 947-1-4 · Zbl 1219.91143 |
| [5] | Kadomtsev, B. B.; Petviashvili, V. I., Sov. Phys. Dokl., 15, 6, 539-541 (1970) · Zbl 0217.25004 |
| [6] | Ablowitz, M. J.; Segur, H., J. Fluid Mech., 92, 691-715 (1979) · Zbl 0413.76009 |
| [7] | Pelinovsky, D. E.; Stepanyants, Y. A.; Kivshar, Y. A., Phys. Rev. E, 51, 5016-5026 (1995) |
| [8] | Dryuma, V. S., Pisma Zh. Eksp. Teor. Fiz., 19, 12, 219-225 (1973) |
| [9] | Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press · Zbl 0762.35001 |
| [10] | Matveev, V. B., Lett. Math. Phys., 3, 213-216 (1979) · Zbl 0418.35005 |
| [11] | Dubrovin, B. A., Russian Math. Surveys, 36, 2, 11-92 (1981) · Zbl 0549.58038 |
| [12] | Freeman, N. C.; Nimmo, J. J.C., Phys. Lett., 96 A, 9, 443-446 (1983) |
| [13] | Freeman, N. C.; Nimmo, J. J.C., J. Phys. A, 17 A, 1415-1424 (1984) · Zbl 0552.35071 |
| [14] | Prinari, B., Inverse scattering theory for the KP equations (1999), (thesis) |
| [15] | Manakov, S. V.; Zakharov, V. E.; Bordag, L. A.; Matveev, V. B., Phys. Lett., 63 A, 3, 205-206 (1977) |
| [16] | Krichever, I., Funct. Anal. and Appl., 12, 1, 76-78 (1978) · Zbl 0374.70008 |
| [17] | Krichever, I.; Novikov, S. P., Funkt. Ana. E Pril., 12, 41-52 (1979) · Zbl 0393.35061 |
| [18] | Satsuma, J.; Ablowitz, M. J., J. Math. Phys., 20, 1496-1503 (1979) · Zbl 0422.35014 |
| [19] | Pelinovsky, D. E.; Stepanyants, Y. A., Phys. JETP Lett., 57, 24-28 (1993) |
| [20] | Pelinovsky, D. E., J. Math. Phys., 35, 5820-5830 (1994) · Zbl 0817.35097 |
| [21] | Ablowitz, M. J.; Villarroel, J., Phys. Rev. Lett., 78, 570-573 (1997) · Zbl 0944.81013 |
| [22] | Villarroel, J.; Ablowitz, M. J., Comm. Math. Phys., 207, 1-42 (1999) · Zbl 0947.35145 |
| [23] | Biondini, G.; Kodama, Y., J. Phys. A: Math. Gen., 36, 10519-10536 (2003) · Zbl 1116.37316 |
| [24] | Kodama, Y., J. Phys. A: Math. Gen., 37, 11169-11190 (2004) · Zbl 1086.35093 |
| [25] | Biondini, G., Phys. Rev. lett., 99 (2007), 064103-1-4 |
| [26] | Gaillard, P., J. Phys. A, 44, 1-15 (2010) |
| [27] | Gaillard, P., J. Math. Sci. Adv. Appl., 13, 2, 71-153 (2012) · Zbl 1260.35202 |
| [28] | Gaillard, P., J. Math. Phys., 54 (2013), 013504-1-32 |
| [29] | Gaillard, P., Adv. Res., 4, 346-364 (2015) |
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