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

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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

Abstract
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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.

Keywords: long-time asymptotics., finite-difference schemes, Weak attractors for dispersive equations.

Mathematics Subject Classification: 35B40, 35B41, 35C08, 37K40, 37L30, 65M06, 65M12, 65N06, 81Q0.

Citation: Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete & Continuous Dynamical Systems - A, 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, (1992).
[2]	J. Chabassier and P. Joly, Energy preserving schemes for nonlinear hamiltonian systems of wave equations. Application to the vibrating piano string,, (2010), (2010). 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. 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, (2002).
[5]	M. S. Èskina, The scattering problem for partial-difference equations,, in Math. Phys. No. 3 (1967) (Russian), (1967), 248.
[6]	C. Foias, M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Dissipativity of numerical schemes,, Nonlinearity, 4 (1991), 591.
[7]	C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds,, Nonlinearity, 4 (1991), 135.
[8]	D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property,, J. Comput. Appl. Math., 134 (2001), 37. doi: 10.1016/S0377-0427(00)00527-6.
[9]	L. Hörmander, "The Analysis of Linear Partial Differential Operators. I,", Springer Study Edition, (1990).
[10]	S. Jiménez and L. Vázquez, Analysis of four numerical schemes for a nonlinear Klein-Gordon equation,, Appl. Math. Comput., 35 (1990), 61.
[11]	L. V. Kapitanskiĭ and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations,, Algebra i Analiz, 2 (1990), 114.
[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. 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.
[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. 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).
[16]	E. A. Kopylova, Dispersive estimates for discrete Schrödinger and Klein-Gordon equations,, Algebra i Analiz, 21 (2009), 87. doi: 10.1090/S1061-0022-2010-01115-4.
[17]	B. Y. Levin, "Lectures on Entire Functions,", vol. 150 of Translations of Mathematical Monographs, (1996).
[18]	J.-L. Lions, Supports de produits de composition. I,, C. R. Acad. Sci. Paris, 232 (1951), 1530.
[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. 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.
[21]	I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction,, Bull. Soc. Math. France, 91 (1963), 129.
[22]	I. E. Segal, Non-linear semi-groups,, Ann. of Math. (2), 78 (1963), 339.
[23]	A. Soffer, Soliton dynamics and scattering,, in, (2006), 459.
[24]	W. A. Strauss, Decay and asymptotics for $\square u = f(u)$,, J. Functional Analysis, 2 (1968), 409.
[25]	W. Strauss and L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation,, J. Comput. Phys., 28 (1978), 271.
[26]	W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators,, Appl. Anal., 80 (2001), 525. 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.
[28]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", vol. 68 of Applied Mathematical Sciences, (1997).
[29]	E. Titchmarsh, The zeros of certain integral functions,, Proc. of the London Math. Soc., 25 (1926), 283.
[30]	J. Virieux, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method,, Geophysics, 51 (1986), 889.
[31]	K. Yosida, "Functional Analysis,", vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1980).

show all references

References:

[1]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", vol. 25 of Studies in Mathematics and its Applications, (1992).
[2]	J. Chabassier and P. Joly, Energy preserving schemes for nonlinear hamiltonian systems of wave equations. Application to the vibrating piano string,, (2010), (2010). 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. 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, (2002).
[5]	M. S. Èskina, The scattering problem for partial-difference equations,, in Math. Phys. No. 3 (1967) (Russian), (1967), 248.
[6]	C. Foias, M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Dissipativity of numerical schemes,, Nonlinearity, 4 (1991), 591.
[7]	C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds,, Nonlinearity, 4 (1991), 135.
[8]	D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property,, J. Comput. Appl. Math., 134 (2001), 37. doi: 10.1016/S0377-0427(00)00527-6.
[9]	L. Hörmander, "The Analysis of Linear Partial Differential Operators. I,", Springer Study Edition, (1990).
[10]	S. Jiménez and L. Vázquez, Analysis of four numerical schemes for a nonlinear Klein-Gordon equation,, Appl. Math. Comput., 35 (1990), 61.
[11]	L. V. Kapitanskiĭ and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations,, Algebra i Analiz, 2 (1990), 114.
[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. 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.
[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. 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).
[16]	E. A. Kopylova, Dispersive estimates for discrete Schrödinger and Klein-Gordon equations,, Algebra i Analiz, 21 (2009), 87. doi: 10.1090/S1061-0022-2010-01115-4.
[17]	B. Y. Levin, "Lectures on Entire Functions,", vol. 150 of Translations of Mathematical Monographs, (1996).
[18]	J.-L. Lions, Supports de produits de composition. I,, C. R. Acad. Sci. Paris, 232 (1951), 1530.
[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. 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.
[21]	I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction,, Bull. Soc. Math. France, 91 (1963), 129.
[22]	I. E. Segal, Non-linear semi-groups,, Ann. of Math. (2), 78 (1963), 339.
[23]	A. Soffer, Soliton dynamics and scattering,, in, (2006), 459.
[24]	W. A. Strauss, Decay and asymptotics for $\square u = f(u)$,, J. Functional Analysis, 2 (1968), 409.
[25]	W. Strauss and L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation,, J. Comput. Phys., 28 (1978), 271.
[26]	W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators,, Appl. Anal., 80 (2001), 525. 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.
[28]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", vol. 68 of Applied Mathematical Sciences, (1997).
[29]	E. Titchmarsh, The zeros of certain integral functions,, Proc. of the London Math. Soc., 25 (1926), 283.
[30]	J. Virieux, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method,, Geophysics, 51 (1986), 889.
[31]	K. Yosida, "Functional Analysis,", vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1980).

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