On Ground-Traveling Waves for the Generalized Kadomtsev-Petviashvili Equations
Tóm tắt
As a continuation of our previous work, we improve some results on convergence of periodic KP traveling waves to solitary ones as the period goes to infinity. In addition, we present some qualitative properties of such waves, as well as nonexistence results, in the case of general nonlinearities. We suggest an approach which does not use any scaling argument.
Tài liệu tham khảo
Ablowitz, M. J., Segur, H. and Wang, X. P.: Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation, Physica D 78(3-4) (1994), 241–265.
Bartsch, Th. and Willem, M.: Infinitely many radial solutions of a semilinear elliptic problem on ?N, Arch. Rat. Mech. Anal. 124 (1993), 261–276.
Bourgin, J.: On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal. 3(4) (1993), 315–341.
De Bouard, A. and Saut, J.-C.: Remarks on the stability of generalized KP solitary waves, Contemp. Math. 200 (1996), 75–84.
De Bouard, A. and Saut, J.-C.: Solitary waves of generalized Kadomtsev-Petviashvili equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 211–236.
De Bouard, A. and Saut, J.-C.: Symmetry and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal. 28 (1997), 1064–1085.
Kadomtsev, B. B. and Petviashvili, V. I.: On stability of waves in weakly dispersive media, Soviet Phys. Dokl. 15 (1970), 539–541, transl. from Dokl. AN SSSR 192 (1970), 753-756.
Krichever, I. M. and Novikov, S. P.: Holomorphic bundles over algebraic curves and nonlinear equations, Russ.Math. Surv. 35(6) (1980), 53–79, transl. from Uspekhi Mat. Nauk 35(6) (1980), 47-68.
Lions, P.-L.: The concentration-compactness method in the calculus of variations. The locally compact case. I, II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145, 223-283.
Liu Yue and Wang, X. P.: Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation, Comm. Math. Phys. 183 (1997), 253–266.
Lizorkin, P. I.: Multipliers of Fourier integrals, Proc. Steklov Inst. Math. 89 (1967), 269–290.
Lopes, O.: A constrained minimization problem with integrals on the entier space, Bol. Soc. Brasil Mat. (N.S.) 25 (1994), 77–92.
Nehari, Z.: On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101–123.
Pankov, A. A.: Semilinear elliptic equations in Rn with nonstabilizing coefficients, Ukrainian Math. J. 41(9) (1989), 1075–1078, transl. from Ukrain. Mat. Zh. 41(9) (1989), 1247-1251.
Pankov, A. A.: On positive solutions of nonlinear elliptic equations on whole space, Soviet Math. Dokl. 44 (1991), 337–341, transl. from Dokl. AN SSSR 319(6) (1991), 1318-1321.
Pankov, A. A. and Pflüger, K.: On a semilinear Schrödinger equation with periodic potential, Nonlinear Anal. 33 (1998), 593–609.
Pankov, A. A. and Pflüger, K.: Periodic and solitary traveling waves for the generalized Kadomtsev-Petviashvili equation, Math. Meth. Appl. Sci. 22 (1999), 733–752.
Pouget, J.: Stability of nonlinear structures in a lattice model for phase transformations in alloys, Phys. Rev. B 46 (1992), 10554–10562.
Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conf. Ser. Math. 65, Amer. Math. Soc., Providence, 1986.
Saut, J.-C.: Remarks on the generalized Kadomtsev-Petviashvili equation, Indiana Univ. Math. J. 42 (1993), 1011–1026.
Stein, E. M. and Weiss, G.: Introduction to Fourier Analysis on Euclidian Spaces, Princeton Univ. Press, Princeton, 1971.
Willem, M.: Minimax Methods, Birkhäuser, Boston, 1996.
Willem, M.: On the generalized Kadomtsev-Petviashvili equation, Rapp. Semin. Math. Louvain Nov. Ser. 245-260 (1996), 213–222.
Zaitsev, A. A.: On formation of nonlinear stationary waves by means of superposition of solitons, Dokl. Akad. Nauk SSSR 272 (1983), 583–587 (in Russian).