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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 |
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. Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
[30] |
J. Virieux, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method,, Geophysics, 51 (1986), 889. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
[30] |
J. Virieux, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method,, Geophysics, 51 (1986), 889. Google Scholar |
[31] |
K. Yosida, "Functional Analysis,", vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1980).
|
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