Global well-posedness for the KP-II equation on the background of a non-localized solution
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 5, pp. 653-676.

Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable in ×𝕋 and perturbations that are square integrable in 2 . In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.

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     author = {Molinet, Luc and Saut, Jean-Claude and Tzvetkov, Nikolay},
     title = {Global well-posedness for the {KP-II} equation on the background of a non-localized solution},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {653--676},
     publisher = {Elsevier},
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Molinet, Luc; Saut, Jean-Claude; Tzvetkov, Nikolay. Global well-posedness for the KP-II equation on the background of a non-localized solution. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 5, pp. 653-676. doi : 10.1016/j.anihpc.2011.04.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2011.04.004/

[1] M.J. Ablowitz, J. Villarroel, The Cauchy problem for the Kadomtsev–Petviashvili II equation with non-decaying data along a line, Stud. Appl. Math. 109 (2002), 151-162 | MR | Zbl

[2] M.J. Ablowitz, J. Villarroel, The Cauchy problem for the Kadomtsev–Petviashvili II equation with data that do not decay along a line, Nonlinearity 17 (2004), 1843-1866 | MR | Zbl

[3] A. De Bouard, J.-C. Saut, Solitary waves of the generalized KP equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 no. 2 (1997), 211-236 | EuDML | Numdam | MR | Zbl

[4] J. Bourgain, On the Cauchy problem for the Kadomtsev–Petviashvili equation, GAFA 3 (1993), 315-341 | EuDML | MR | Zbl

[5] P.G. Drazin, R.S. Johnson, Solitons: An Introduction, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge (1989) | MR | Zbl

[6] A.S. Fokas, A.K. Pogorobkov, Inverse scattering transform for the KP-I equation on the background of a one-line soliton, Nonlinearity 16 (2003), 771-783 | MR

[7] J. Ginibre, Le problème de Cauchy pour des EDP semi linéaires périodiques en variables dʼespaces, Sém. Bourbaki (1994–1995), 163-187 | EuDML | Numdam

[8] A. Grünrock, M. Panthee, J. Drumond Silva, On KP II equations on cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 6 (2009), 2335-2358 | EuDML | Numdam | MR | Zbl

[9] M. Hadac, Well-posedness for the Kadomtsev (KP II) equation and generalizations, Trans. Amer. Math. Soc. 360 (2008), 6555-6572 | MR | Zbl

[10] M. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 3 (2009), 917-941 | EuDML | Numdam | MR | Zbl

[11] A.D. Ionescu, C. Kenig, D. Tataru, Global well-posedness of the initial value problem for the KP I equation in the energy space, Invent. Math. 173 no. 2 (2008), 265-304 | MR | Zbl

[12] P. Isaza, J. Mejia, Local and global Cauchy problems for the Kadomtsev–Petviashvili (KP-II) equation in Sobolev spaces of negative indices, Comm. Partial Differential Equations 26 (2001), 1027-1057 | MR | Zbl

[13] B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Soviet Phys. Dokl. 15 no. 6 (1970), 539-541 | Zbl

[14] C. Kenig, On the local and global well-posedness for the KP-I equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 827-838 | EuDML | Numdam | MR | Zbl

[15] T. Mizumachi, N. Tzvetkov, Stability of the line soliton of the KP II equation under periodic transverse perturbations, arXiv:1008.0812v1 | MR | Zbl

[16] L. Molinet, On the asymptotic behavior of solutions to the (generalized) Kadomtsev–Petviashvili–Burgers equation, J. Differential Equations 152 (1999), 30-74 | MR | Zbl

[17] L. Molinet, J.-C. Saut, N. Tzvetkov, Global well-posedness for the KP-I equation, Math. Ann. 324 (2002), 255-275, Math. Ann. 328 (2004), 707-710 | MR | Zbl

[18] L. Molinet, J.-C. Saut, N. Tzvetkov, Global well-posedness of the KP-I equation on the background of a non localized solution, Comm. Math. Phys. 272 (2007), 775-810 | MR | Zbl

[19] S. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Contemp. Soviet Math., Consultants Bureau, New York, London (1984) | MR | Zbl

[20] F. Rousset, N. Tzvetkov, Transverse instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 477-496 | EuDML | Numdam | MR | Zbl

[21] F. Rousset, N. Tzvetkov, Transverse nonlinear instability for some Hamiltonian PDEʼs, J. Math. Pures Appl. 90 (2008), 550-590 | MR | Zbl

[22] J. Satsuma, N-soliton of the two-dimensional Korteweg–de Vries equation, J. Phys. Soc. Japan 40 (1976), 286-290 | MR | Zbl

[23] J.-C. Saut, Remarks on the generalized Kadomtsev–Petviashvili equations, Indiana Univ. Math. J. 42 (1993), 1017-1029 | MR

[24] J.-C. Saut, N. Tzvetkov, On the periodic KP-I type equations, Comm. Math. Phys. 221 (2001), 451-476 | MR | Zbl

[25] J.-C. Saut, N. Tzvetkov, The Cauchy problem for higher order KP equations, J. Differential Equations 153 no. 1 (1999), 196-222 | MR | Zbl

[26] J.-C. Saut, N. Tzvetkov, The Cauchy problem for the fifth order KP equation, J. Math. Pures Appl. 79 no. 4 (2000), 307-338 | MR | Zbl

[27] H. Takaoka, Global well-posedness for the Kadomtsev–Petviashvili II equation, Discrete Contin. Dyn. Syst. 6 (2000), 483-499 | MR | Zbl

[28] H. Takaoka, N. Tzvetkov, On the local regularity of Kadomtsev–Petviashvili-II equation, IMRN 8 (2001), 77-114 | MR | Zbl

[29] N. Tzvetkov, Global low regularity solutions for Kadomtsev–Petviashvili equation, Differential Integral Equations 13 (2000), 1289-1320 | MR | Zbl

[30] V.E. Zakharov, Instability and nonlinear oscillations of solitons, JETP Lett. 22 (1975), 172-173

[31] X. Zhou, Inverse scattering transform for the time dependent Schrödinger equation with applications to the KP-I equation, Comm. Math. Phys. 128 (1990), 551-564 | MR | Zbl

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