Solitary waves for nonlinear Klein–Gordon equations coupled with Born–Infeld theory

Yu, Yong

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 351-376.

We consider the nonlinear Klein–Gordon equations coupled with the Born–Infeld theory under the electrostatic solitary wave ansatz. The existence of the least-action solitary waves is proved in both bounded smooth domain case and ${ℝ}^3$ case. In particular, for bounded smooth domain case, we study the asymptotic behaviors and profiles of the positive least-action solitary waves with respect to the frequency parameter ω. We show that when κ and ω are suitably large, the least-action solitary waves admit only one local maximum point. When $\omega \to \infty$, the point-condensation phenomenon occurs if we consider the normalized least-action solitary waves.

[1] J.C. Brunelli, Dispersionless limit of integrable models, Brazilian J. Physics 30 no. 2 (June 2000), 455-468

[2] Max Born, Modified field equations with a finite radius of the electron, Nature 132 (1933), 282 | Zbl

[3] Max Born, On the quantum theory of the electromagnetic field, Proc. Roy. Soc. A 143 (1934), 410-437 | Zbl

[4] V. Benci, D. Fortunato, Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations, Rev. Math. Phys. 14 no. 4 (2002), 409-420 | MR | Zbl

[5] V. Benci, D. Fortunato, Solitary waves in the nonlinear wave equation and in gauge theories, Fixed Point Theory Appl. 1 (2007), 61-86 | MR | Zbl

[6] V. Benci, D. Fortunato, Solitary waves in classical field theory, V. Benci, A. Masiello (ed.), Nonlinear Analysis and Applications to Physical Sciences, Springer, Milano (2004), 1-50 | MR | Zbl

[7] V. Benci, D. Fortunato, On the existence of infinitely many geodesics on space–time manifolds, Adv. in Math. 105 (1994), 1-25 | MR | Zbl

[8] M. Born, L. Infeld, Foundation of the new field theory, Nature 132 (1933), 1004 | Zbl

[9] M. Born, L. Infeld, Foundation of the new field theory, Proc. Roy. Soc. A 144 (1934), 425-451 | Zbl

[10] V. Benci, P.H. Rabinowitz, Critical points theorems for indefinite functionals, Invent. Math. 52 (1979), 241-273 | EuDML | MR | Zbl

[11] D. Cassani, Existence and non-existence of solitary waves for the critical Klein–Gordon equation coupled with Maxwell's equations, Nonlinear Anal. 58 (2004), 733-747 | MR | Zbl

[12] Teresa D'Aprile, Dimitri Mugnai, Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 no. 5 (2004), 893-906 | MR | Zbl

[13] Pietro D'Avenia, Lorenzo Pisani, Nonlinear Klein–Gordon equations coupled with Born–Infeld type equations, Elect. J. Diff. Eqns. 26 (2002), 1-13 | EuDML | MR | Zbl

[14] H. Egnell, Asymptotic results for finite energy solutions of semilinear elliptic equations, J. Differential Equations 98 (1992), 34-56 | MR | Zbl

[15] M. Esteban, P.L. Lions, Existence and non-existence results for semilinear elliptic problems in unbounded domanis, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982), 1-14 | MR | Zbl

[16] M. Esteban, E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys. 171 (1995), 323-350 | MR | Zbl

[17] D. Fortunato, L. Orsina, L. Pisani, Born–Infeld type equations for electrostatic fields, J. of Math. Phys. 43 no. 11 (2002), 5698-5706 | MR | Zbl

[18] G.W. Gibbons, Born–Infeld particles and Dirichlet p-branes, Nucl. Phys. B 514 (1998), 603 | MR | Zbl

[19] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Adv. in Math. 7A no. Suppl. Stud. (1981), 369-402 | MR

[20] D. Gilbarg, Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer (2001) | MR | Zbl

[21] M.K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal. 105 (1989), 243-266 | MR | Zbl

[22] E. Long, Existence and stability of solitary waves in non-linear Klein–Gordon–Maxwell equations, Rev. Math. Phys. 18 (2006), 747-779 | MR | Zbl

[23] C.-S. Lin, W.-M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), 1-27 | MR | Zbl

[24] F. Lin, Y. Yang, Gauged harmonic maps, Born–Infeld electromagnetism, and magnetic vortices, CPAM 56 (2003), 1631-1665 | MR | Zbl

[25] D. Mugnai, Coupled Klein–Gordon and Born–Infeld type equations: Looking for solitary waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), 1519-1528 | MR | Zbl

[26] W.-M. Ni, J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, CPAM XLVIII (1995), 731-768 | MR | Zbl

[27] N. Ogawa, Chaplygin Gas and Brane, Proceedings of the 8th International Conference on Geometry Integrability & Quantization (June 2007), 279-291 | MR

[28] R.S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), 19-30 | MR | Zbl

[29] J. Polchinski, TASI lectures on D-branes, arXiv:hep-th/9611050 R. Argurio, Brane physics in M-theory, hep-th/9807171 K.G. Savvidy, Born–Infeld action in string theory, hep-th/9906075

[30] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Reg. Conf. Ser. Math. vol. 65 (1986) | MR

[31] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edition

[32] N. Seilberg, E. Witten, String theory and noncommutative geometry, JHEP 9909 (1999), 032 | MR

[33] W.-M. Ni, I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, CPAM 44 (1991), 819-851 | MR | Zbl

[34] Y. Yang, Classical solutions in the Born–Infeld theory, Proceedings: Mathematical, Physical and Engineering Sciences 456 no. 1995 (2000), 615-640 | MR | Zbl

[35] X.-P. Zhu, Multiple entire solutions of a semilinear elliptic equation, Nonlinear Anal. 12 (1988), 1297-1316 | MR | Zbl

[36] Z. Zhang, K. Li, Spike-layered solutions of singularly perturbed quasilinear Dirichlet problems, J. Math. Anal. Appl. 283 (2003), 667-680 | MR | Zbl