# American Institute of Mathematical Sciences

July  2013, 33(7): 2711-2755. doi: 10.3934/dcds.2013.33.2711

## Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator

 1 Texas A&M University, College Station, Texas 77843, United States

Received  April 2012 Revised  October 2012 Published  January 2013

We consider the $\mathbf{U}(1)$-invariant nonlinear Klein-Gordon equation in discrete space and discrete time, which is the discretization of the nonlinear continuous Klein-Gordon equation. To obtain this equation, we use the energy-conserving finite-difference scheme of Strauss-Vazquez. We prove that each finite energy solution converges as $T → ± ∞$ to the finite-dimensional set of all multifrequency solitary wave solutions with one, two, and four frequencies. The components of the solitary manifold corresponding to the solitary waves of the first two types are generically two-dimensional, while the component corresponding to the last type is generically four-dimensional. The attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent radiation. For the proof, we develop the well-posedness for the nonlinear wave equation in discrete space-time, apply the technique of quasimeasures, and also obtain the version of the Titchmarsh convolution theorem for distributions on the circle.
Citation: Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2711-2755. doi: 10.3934/dcds.2013.33.2711
##### References:
 [1] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," vol. 25 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1992, translated and revised from the 1989 Russian original by Babin. [2] J. Chabassier and P. Joly, Energy preserving schemes for nonlinear hamiltonian systems of wave equations. Application to the vibrating piano string, (2010), to appear in Comp. Methods in Appl. Mech. and Engineering. doi: 10.1016/j.cma.2010.04.013. [3] A. Comech and A. I. Komech, Well-posedness and the energy and charge conservation for nonlinear wave equations in discrete space-time, Russ. J. Math. Phys., 18 (2011), 410-419. doi: 10.1134/S1061920811040030. [4] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002. [5] M. S. Èskina, The scattering problem for partial-difference equations, in Math. Phys. No. 3 (1967) (Russian), 248-273, Naukova Dumka, Kiev, 1967. [6] C. Foias, M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613. [7] C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153. [8] D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, J. Comput. Appl. Math., 134 (2001), 37-57. doi: 10.1016/S0377-0427(00)00527-6. [9] L. Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer Study Edition, Springer-Verlag, Berlin, 1990, second edn. [10] S. Jiménez and L. Vázquez, Analysis of four numerical schemes for a nonlinear Klein-Gordon equation, Appl. Math. Comput., 35 (1990), 61-94. [11] L. V. Kapitanskiĭ and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Algebra i Analiz, 2 (1990), 114-140. [12] A. I. Komech and A. A. Komech, Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field, Arch. Ration. Mech. Anal., 185 (2007), 105-142. doi: 10.1007/s00205-006-0039-z. [13] A. I. Komech and A. A. Komech, Global attraction to solitary waves for Klein-Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2008), 855-868. [14] A. I. Komech and A. A. Komech, Global attractor for the Klein-Gordon field coupled to several nonlinear oscillators, J. Math. Pures Appl., 93 (2010), 91-111. doi: 10.1016/j.matpur.2009.08.011. [15] A. I. Komech and A. A. Komech, On the Titchmarsh convolution theorem for distributions on a circle, Funktsional. Anal. i Prilozhen., 46 (2012), to appear (see arXiv:1108.2463). [16] E. A. Kopylova, Dispersive estimates for discrete Schrödinger and Klein-Gordon equations, Algebra i Analiz, 21 (2009), 87-113. doi: 10.1090/S1061-0022-2010-01115-4. [17] B. Y. Levin, "Lectures on Entire Functions," vol. 150 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996, in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko. [18] J.-L. Lions, Supports de produits de composition. I, C. R. Acad. Sci. Paris, 232 (1951), 1530-1532. [19] S. Li and L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 32 (1995), 1839-1875. doi: 10.1137/0732083. [20] C. S. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math., 25 (1972), 1-31. [21] I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), 129-135. [22] I. E. Segal, Non-linear semi-groups, Ann. of Math. (2), 78 (1963), 339-364. [23] A. Soffer, Soliton dynamics and scattering, in "International Congress of Mathematicians. Vol. III," 459-471, Eur. Math. Soc., Zürich, 2006. [24] W. A. Strauss, Decay and asymptotics for $\square u = f(u)$, J. Functional Analysis, 2 (1968), 409-457. [25] W. Strauss and L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys., 28 (1978), 271-278. [26] W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators, Appl. Anal., 80 (2001), 525-556. doi: 10.1080/00036810108841007. [27] T. Tao, A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations, Dyn. Partial Differ. Equ., 4 (2007), 1-53. [28] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," vol. 68 of Applied Mathematical Sciences, Springer-Verlag, New York, 1997, second edn. [29] E. Titchmarsh, The zeros of certain integral functions, Proc. of the London Math. Soc., 25 (1926), 283-302. [30] J. Virieux, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method, Geophysics, 51 (1986), 889-901. [31] K. Yosida, "Functional Analysis," vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1980, sixth edn.

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##### References:
 [1] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," vol. 25 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1992, translated and revised from the 1989 Russian original by Babin. [2] J. Chabassier and P. Joly, Energy preserving schemes for nonlinear hamiltonian systems of wave equations. Application to the vibrating piano string, (2010), to appear in Comp. Methods in Appl. Mech. and Engineering. doi: 10.1016/j.cma.2010.04.013. [3] A. Comech and A. I. Komech, Well-posedness and the energy and charge conservation for nonlinear wave equations in discrete space-time, Russ. J. Math. Phys., 18 (2011), 410-419. doi: 10.1134/S1061920811040030. [4] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002. [5] M. S. Èskina, The scattering problem for partial-difference equations, in Math. Phys. No. 3 (1967) (Russian), 248-273, Naukova Dumka, Kiev, 1967. [6] C. Foias, M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613. [7] C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153. [8] D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, J. Comput. Appl. Math., 134 (2001), 37-57. doi: 10.1016/S0377-0427(00)00527-6. [9] L. Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer Study Edition, Springer-Verlag, Berlin, 1990, second edn. [10] S. Jiménez and L. Vázquez, Analysis of four numerical schemes for a nonlinear Klein-Gordon equation, Appl. Math. Comput., 35 (1990), 61-94. [11] L. V. Kapitanskiĭ and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Algebra i Analiz, 2 (1990), 114-140. [12] A. I. Komech and A. A. Komech, Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field, Arch. Ration. Mech. Anal., 185 (2007), 105-142. doi: 10.1007/s00205-006-0039-z. [13] A. I. Komech and A. A. Komech, Global attraction to solitary waves for Klein-Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2008), 855-868. [14] A. I. Komech and A. A. Komech, Global attractor for the Klein-Gordon field coupled to several nonlinear oscillators, J. Math. Pures Appl., 93 (2010), 91-111. doi: 10.1016/j.matpur.2009.08.011. [15] A. I. Komech and A. A. Komech, On the Titchmarsh convolution theorem for distributions on a circle, Funktsional. Anal. i Prilozhen., 46 (2012), to appear (see arXiv:1108.2463). [16] E. A. Kopylova, Dispersive estimates for discrete Schrödinger and Klein-Gordon equations, Algebra i Analiz, 21 (2009), 87-113. doi: 10.1090/S1061-0022-2010-01115-4. [17] B. Y. Levin, "Lectures on Entire Functions," vol. 150 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996, in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko. [18] J.-L. Lions, Supports de produits de composition. I, C. R. Acad. Sci. Paris, 232 (1951), 1530-1532. [19] S. Li and L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 32 (1995), 1839-1875. doi: 10.1137/0732083. [20] C. S. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math., 25 (1972), 1-31. [21] I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), 129-135. [22] I. E. Segal, Non-linear semi-groups, Ann. of Math. (2), 78 (1963), 339-364. [23] A. Soffer, Soliton dynamics and scattering, in "International Congress of Mathematicians. Vol. III," 459-471, Eur. Math. Soc., Zürich, 2006. [24] W. A. Strauss, Decay and asymptotics for $\square u = f(u)$, J. Functional Analysis, 2 (1968), 409-457. [25] W. Strauss and L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys., 28 (1978), 271-278. [26] W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators, Appl. Anal., 80 (2001), 525-556. doi: 10.1080/00036810108841007. [27] T. Tao, A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations, Dyn. Partial Differ. Equ., 4 (2007), 1-53. [28] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," vol. 68 of Applied Mathematical Sciences, Springer-Verlag, New York, 1997, second edn. [29] E. Titchmarsh, The zeros of certain integral functions, Proc. of the London Math. Soc., 25 (1926), 283-302. [30] J. Virieux, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method, Geophysics, 51 (1986), 889-901. [31] K. Yosida, "Functional Analysis," vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1980, sixth edn.
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