Abstract
We prove that the solution of the Cauchy problem for the Kadomtsev–Petviashvili-I Equation obtained by the inverse spectral method belongs to the Sobolev space Hk(R2) for k ≥ 0, under the assumption that the initial datum is a small Schwartz function. This solution is shown to be the unique solution within a class of generalized solutions of the Kadomtsev–Petviashvili-I equation.
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Ablowitz, M. J. and Villarroel, J.: On the Kadomtsev–Petviashvili equation and associated constraints, Stud. Appl. Math. 85 (1991), 195–213.
Boiti, M., Pempinelli, F. and Pogrebkov, A.: Solutions of the KPI equation with smooth initial data, Inverse Problems 10 (1994), 505–519.
Boiti, M., Pempinelli, F. and Pogrebkov, A.: Properties of solutions of the Kadomtsev–Petviashvili I equation, J. Math. Phys. 35 (1994), 4683–4718.
Faminskii, A. V.: The Cauchy problem for the Kadomtsev–Petviashvili equation, Russian Math. Surveys 5 (1990), 203–204.
Fokas, A. S. and Ablowitz, M. J.: On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev–Petviashvili (I) equation, Stud. Appl. Math. 69 (1983), 211–228.
Fokas, A. S. and Sung, L.-Y.: The Cauchy problem for the Kadomtsev–Petviashvili-I equation without the zero mass constraint, Math. Proc. Camb. Phil. Soc. 125 (1999), 113–138.
Johnson, R. S.: Water waves and Korteweg–de Vries equations, Fluid Mech. 97 (1980), 701–719.
Kadomtsev, B. B. and Petviashvili, V. I.: On the stability of solitary waves in weakly dispersive media, Soviet Phys. Dokl. 15 (1970), 539–541.
Lipovskij, V. D., Matveev, V. B. and Smirnov, A. O.: On the isomorphism between KP-equation and Johnson equation, Zap. Semin. LOMI 150 (1986), 70–75.
Manakov, S. V.: The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev–Petviashvili equation, Physica D 3 (1981), 420–427.
Muskhelishvili, N. I.: Singular Integral Equations, Dover, New York, 1992.
Saut, J.-C.: Remarks on the generalized Kadomtsev–Petviashvili equations, Indiana Math. J. 42 (1993), 1011–1027.
Saut, J.-C.: Recent results on the generalized Kadomtsev–Petviashvili equations, Acta Appl. Math. 39 (1995), 477–487.
Segur, H.: Comments on IS for the Kadomtsev–Petviashvili equation, in: M. Tabor and Y. M. Treve (eds.), Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, AIP Conference Proceedings, No. 88, 1982, pp. 211–228.
Stein, E.: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
Tom, M. M.: On a generalized Kadomtsev–Petviashvili equation, Contemp. Math. 200 (1996), 193–210.
Ukai, S.: Local solutions of the Kadomtsev–Petviashvili equation, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 36 (1989), 193–209.
Wloka, J. T.: Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
Yosida, K.: Functional Analysis, Springer-Verlag, Berlin, 1995.
Zhou, X.: Inverse scattering transform for the time dependent Schrödinger equation with applications to the KP-I equation, Comm. Math. Phys. 128 (1990), 551–564.
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Sung, Ly. Square Integrability and Uniqueness of the Solutions of the Kadomtsev–Petviashvili-I Equation. Mathematical Physics, Analysis and Geometry 2, 1–24 (1999). https://doi.org/10.1023/A:1009806923447
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DOI: https://doi.org/10.1023/A:1009806923447