Finite mechanical proxies for a class of reducible continuum systems

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September 2014, 9(3): 417-432. doi: 10.3934/nhm.2014.9.417

Finite mechanical proxies for a class of reducible continuum systems

Franco Cardin ^1, and Alberto Lovison ^1,

1.	Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63 - 35121 Padova, Italy, Italy

Received June 2013 Revised May 2014 Published October 2014

Abstract
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We present the exact finite reduction of a class of nonlinearly perturbed wave equations --typically, a non-linear elastic string-- based on the Amann--Conley--Zehnder paradigm. By solving an inverse eigenvalue problem, we establish an equivalence between the spectral finite description derived from A--C--Z and a discrete mechanical model, a well definite finite spring--mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem.

Keywords: thermodynamical limit, exact finite reductions, collective variables, Nonlinear wave equation, inverse eigenvalue problems..

Mathematics Subject Classification: Primary: 74B20, 70J50; Secondary: 65F1.

Citation: Franco Cardin, Alberto Lovison. Finite mechanical proxies for a class of reducible continuum systems. Networks and Heterogeneous Media, 2014, 9 (3) : 417-432. doi: 10.3934/nhm.2014.9.417

References:

[1]	H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 539-603.
[2]	H. Amann and E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations, Houston J. Math., 7 (1981), 147-174.
[3]	H. Amann, Multiple positive fixed points of asymptotically linear maps, J. Functional Analysis, 17 (1974), 174-213. doi: 10.1016/0022-1236(74)90011-1.
[4]	H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), 127-166. doi: 10.1007/BF01215273.
[5]	A. Ambrosetti, Critical points and nonlinear variational problems, Mém. Soc. Math. France (N.S.), (1992), 139 pp.
[6]	S. S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370. doi: 10.2307/2321203.
[7]	A. Arsie and C. Ebenbauer, Locating omega-limit sets using height functions, Journal of Differential Equations, 248 (2010), 2458-2469. doi: 10.1016/j.jde.2009.11.012.
[8]	D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69-87. doi: 10.1007/s003320010010.
[9]	J. Berkovits, H. Leinfelder and V. Mustonen, Existence and multiplicity results for wave equations with time-independent nonlinearity, Topol. Methods Nonlinear Anal., 22 (2003), 273-295.
[10]	M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328. doi: 10.1007/s00220-003-0972-8.
[11]	C. Boldrighini, A. De Masi, A. Pellegrinotti and E. Presutti, Collective phenomena in interacting particle systems, Stochastic Process. Appl., 25 (1987), 137-152. doi: 10.1016/0304-4149(87)90194-3.
[12]	H. Brézis, J.-M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math., 33 (1980), 667-684. doi: 10.1002/cpa.3160330507.
[13]	M. Cappiello, Pseudodifferential parametrices of infinite order for SG-hyperbolic problems, Rend. Sem. Mat. Univ. Politec. Torino, 61 (2003), 411-441.
[14]	F. Cardin, Global finite generating functions for field theory, in Classical and quantum integrability (Warsaw, 2001), 59 of Banach Center Publ., 133-142. Polish Acad. Sci., Warsaw, 2003. doi: 10.4064/bc59-0-6.
[15]	F. Cardin and A. Lovison, Microscopic structures from reduction of continuum nonlinear problems, AAPP - Physical, Mathematical, and Natural Sciences, 91 (2013), 1-20. doi: 10.1478/AAPP.91S1A4.
[16]	F. Cardin and C. Tebaldi, Finite reductions for dissipative systems and viscous fluid-dynamic models on $\mathbbT^2$, J. Math. Anal. Appl., 345 (2008), 213-222. doi: 10.1016/j.jmaa.2008.04.012.
[17]	F. Cardin, A. Lovison and M. Putti, Implementation of an exact finite reduction scheme for steady-state reaction-diffusion equations, Internat. J. Numer. Methods Engrg., 69 (2007), 1804-1818. doi: 10.1002/nme.1824.
[18]	H. Chiba, Continuous limit and the moments system for the globally coupled phase oscillators, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1891-1903. doi: 10.3934/dcds.2013.33.1891.
[19]	M. Cicalese, A. DeSimone and C. I. Zeppieri, Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers, Networks and Heterogeneous Media, 4 (2009), 667-708. doi: 10.3934/nhm.2009.4.667.
[20]	C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol'd, Invent. Math., 73 (1983), 33-49. doi: 10.1007/BF01393824.
[21]	C. Conley, Isolated Invariant Sets and The Morse Index, Number 38 in Regional conferences series in mathematics. Conference Board for the Mathematical Sciences, 1976.
[22]	J.-M. Coron, Periodic solutions of a nonlinear wave equation without assumption of monotonicity, Math. Ann., 262 (1983), 273-285. doi: 10.1007/BF01455317.
[23]	R. de la Llave, Variational methods for quasi-periodic solutions of partial differential equations, in Hamiltonian systems and celestial mechanics (Pátzcuaro, 1998), 6 of World Sci. Monogr. Ser. Math., 214-228. World Sci. Publ., River Edge, NJ, 2000. doi: 10.1142/9789812792099_0013.
[24]	M. Degiovanni, On Morse theory for continuous functionals, Conf. Semin. Mat. Univ. Bari, 290 (2003), 1-22.
[25]	A. Di Carlo, private communication.
[26]	C. Ebenbauer and A. Arsie, On an eigenflow equation and its Lie algebraic generalization, Communications in Information and Systems, 8 (2008), 147-170. doi: 10.4310/CIS.2008.v8.n2.a6.
[27]	J. Frenkel and T. Kontorova, On the theory of plastic deformation and twinning, Acad. Sci. U.S.S.R. J. Phys., 1 (1939), 137-149.
[28]	G. M. L. Gladwell, Inverse Problems in Vibration, 119 of Solid Mechanics and its Applications. Kluwer Academic Publishers, Dordrecht, second edition, 2004.
[29]	J. M. Greenberg and A. Nachman, Continuum limits for discrete gases with long- and short-range interactions, Communications on Pure and Applied Mathematics, 47 (1994), 1239-1281. doi: 10.1002/cpa.3160470905.
[30]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine Series 5, 39 (1895), 422-443. doi: 10.1080/14786449508620739.
[31]	A. Lovison, Generating functions and finite parameter reductions in fields theory, Bollettino Della Unione Matematica Italiana, 8A (2005), {569-572}.
[32]	A. Lovison, F. Cardin and A. Bobbo, Discrete structures equivalent to nonlinear Dirichlet and wave equations, Continuum Mech Therm, 21 (2009), 27-40. doi: 10.1007/s00161-009-0097-1.
[33]	M. Lucia, P. Magrone and H.-S. Zhou, A Dirichlet problem with asymptotically linear and changing sign nonlinearity, Rev. Mat. Complut., 16 (2003), 465-481.
[34]	L. Maragliano, A. Fischer, E. Vanden-Eijnden and G. Ciccotti, String method in collective variables: Minimum free energy paths and isocommittor surfaces, Journal of Chemical Physics, 125 (2006), 024106. doi: 10.1063/1.2212942.
[35]	S. Müller and A. Schlömerkemper, Discrete-to-continuum limit of magnetic forces, Comptes Rendus Mathematique, 335 (2002), 393-398. doi: 10.1016/S1631-073X(02)02494-9.
[36]	L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 4 (1981), 267-302. doi: 10.1090/S0273-0979-1981-14888-6.
[37]	P. Nylen and F. Uhlig, Inverse eigenvalue problem: Existence of special spring - mass systems, Inverse Problems, 13 (1997), 1071-1081. doi: 10.1088/0266-5611/13/4/012.
[38]	P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), 145-205. doi: 10.1002/cpa.3160200105.
[39]	P. H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68. doi: 10.1002/cpa.3160310103.
[40]	S. Rybicki, Periodic solutions of vibrating strings. Degree theory approach, Ann. Mat. Pura Appl. (4), 179 (2001), 197-214. doi: 10.1007/BF02505955.
[41]	M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day Inc., San Francisco, Calif., 1964.
[42]	C. Viterbo, Recent progress in periodic orbits of autonomous Hamiltonian systems and applications to symplectic geometry, in Nonlinear functional analysis (Newark, NJ, 1987), 121 of Lecture Notes in Pure and Appl. Math., 227-250. Dekker, New York, 1990.
[43]	V. Volterra, Leçons sur les Fonctions de Ligne, Gauthier-Villars, Paris, 1913.
[44]	J. von Neumann, Proposal and Analysis of a New Numerical Method for the Treatment of Hydrodynamical Shock Problems, AMP Report, (1944), 1-25.
[45]	I. R. Yukhnovskiĭ, Phase Transitions of the Second Order, World Scientific Publishing Co., Singapore, 1987. doi: 10.1142/0289.
[46]	N. J. Zabusky and M. D. Kruskal, Interaction of "solitons'' in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243. doi: 10.1103/PhysRevLett.15.240.

show all references

References:

[1]	H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 539-603.
[2]	H. Amann and E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations, Houston J. Math., 7 (1981), 147-174.
[3]	H. Amann, Multiple positive fixed points of asymptotically linear maps, J. Functional Analysis, 17 (1974), 174-213. doi: 10.1016/0022-1236(74)90011-1.
[4]	H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), 127-166. doi: 10.1007/BF01215273.
[5]	A. Ambrosetti, Critical points and nonlinear variational problems, Mém. Soc. Math. France (N.S.), (1992), 139 pp.
[6]	S. S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370. doi: 10.2307/2321203.
[7]	A. Arsie and C. Ebenbauer, Locating omega-limit sets using height functions, Journal of Differential Equations, 248 (2010), 2458-2469. doi: 10.1016/j.jde.2009.11.012.
[8]	D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69-87. doi: 10.1007/s003320010010.
[9]	J. Berkovits, H. Leinfelder and V. Mustonen, Existence and multiplicity results for wave equations with time-independent nonlinearity, Topol. Methods Nonlinear Anal., 22 (2003), 273-295.
[10]	M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328. doi: 10.1007/s00220-003-0972-8.
[11]	C. Boldrighini, A. De Masi, A. Pellegrinotti and E. Presutti, Collective phenomena in interacting particle systems, Stochastic Process. Appl., 25 (1987), 137-152. doi: 10.1016/0304-4149(87)90194-3.
[12]	H. Brézis, J.-M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math., 33 (1980), 667-684. doi: 10.1002/cpa.3160330507.
[13]	M. Cappiello, Pseudodifferential parametrices of infinite order for SG-hyperbolic problems, Rend. Sem. Mat. Univ. Politec. Torino, 61 (2003), 411-441.
[14]	F. Cardin, Global finite generating functions for field theory, in Classical and quantum integrability (Warsaw, 2001), 59 of Banach Center Publ., 133-142. Polish Acad. Sci., Warsaw, 2003. doi: 10.4064/bc59-0-6.
[15]	F. Cardin and A. Lovison, Microscopic structures from reduction of continuum nonlinear problems, AAPP - Physical, Mathematical, and Natural Sciences, 91 (2013), 1-20. doi: 10.1478/AAPP.91S1A4.
[16]	F. Cardin and C. Tebaldi, Finite reductions for dissipative systems and viscous fluid-dynamic models on $\mathbbT^2$, J. Math. Anal. Appl., 345 (2008), 213-222. doi: 10.1016/j.jmaa.2008.04.012.
[17]	F. Cardin, A. Lovison and M. Putti, Implementation of an exact finite reduction scheme for steady-state reaction-diffusion equations, Internat. J. Numer. Methods Engrg., 69 (2007), 1804-1818. doi: 10.1002/nme.1824.
[18]	H. Chiba, Continuous limit and the moments system for the globally coupled phase oscillators, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1891-1903. doi: 10.3934/dcds.2013.33.1891.
[19]	M. Cicalese, A. DeSimone and C. I. Zeppieri, Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers, Networks and Heterogeneous Media, 4 (2009), 667-708. doi: 10.3934/nhm.2009.4.667.
[20]	C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol'd, Invent. Math., 73 (1983), 33-49. doi: 10.1007/BF01393824.
[21]	C. Conley, Isolated Invariant Sets and The Morse Index, Number 38 in Regional conferences series in mathematics. Conference Board for the Mathematical Sciences, 1976.
[22]	J.-M. Coron, Periodic solutions of a nonlinear wave equation without assumption of monotonicity, Math. Ann., 262 (1983), 273-285. doi: 10.1007/BF01455317.
[23]	R. de la Llave, Variational methods for quasi-periodic solutions of partial differential equations, in Hamiltonian systems and celestial mechanics (Pátzcuaro, 1998), 6 of World Sci. Monogr. Ser. Math., 214-228. World Sci. Publ., River Edge, NJ, 2000. doi: 10.1142/9789812792099_0013.
[24]	M. Degiovanni, On Morse theory for continuous functionals, Conf. Semin. Mat. Univ. Bari, 290 (2003), 1-22.
[25]	A. Di Carlo, private communication.
[26]	C. Ebenbauer and A. Arsie, On an eigenflow equation and its Lie algebraic generalization, Communications in Information and Systems, 8 (2008), 147-170. doi: 10.4310/CIS.2008.v8.n2.a6.
[27]	J. Frenkel and T. Kontorova, On the theory of plastic deformation and twinning, Acad. Sci. U.S.S.R. J. Phys., 1 (1939), 137-149.
[28]	G. M. L. Gladwell, Inverse Problems in Vibration, 119 of Solid Mechanics and its Applications. Kluwer Academic Publishers, Dordrecht, second edition, 2004.
[29]	J. M. Greenberg and A. Nachman, Continuum limits for discrete gases with long- and short-range interactions, Communications on Pure and Applied Mathematics, 47 (1994), 1239-1281. doi: 10.1002/cpa.3160470905.
[30]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine Series 5, 39 (1895), 422-443. doi: 10.1080/14786449508620739.
[31]	A. Lovison, Generating functions and finite parameter reductions in fields theory, Bollettino Della Unione Matematica Italiana, 8A (2005), {569-572}.
[32]	A. Lovison, F. Cardin and A. Bobbo, Discrete structures equivalent to nonlinear Dirichlet and wave equations, Continuum Mech Therm, 21 (2009), 27-40. doi: 10.1007/s00161-009-0097-1.
[33]	M. Lucia, P. Magrone and H.-S. Zhou, A Dirichlet problem with asymptotically linear and changing sign nonlinearity, Rev. Mat. Complut., 16 (2003), 465-481.
[34]	L. Maragliano, A. Fischer, E. Vanden-Eijnden and G. Ciccotti, String method in collective variables: Minimum free energy paths and isocommittor surfaces, Journal of Chemical Physics, 125 (2006), 024106. doi: 10.1063/1.2212942.
[35]	S. Müller and A. Schlömerkemper, Discrete-to-continuum limit of magnetic forces, Comptes Rendus Mathematique, 335 (2002), 393-398. doi: 10.1016/S1631-073X(02)02494-9.
[36]	L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 4 (1981), 267-302. doi: 10.1090/S0273-0979-1981-14888-6.
[37]	P. Nylen and F. Uhlig, Inverse eigenvalue problem: Existence of special spring - mass systems, Inverse Problems, 13 (1997), 1071-1081. doi: 10.1088/0266-5611/13/4/012.
[38]	P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), 145-205. doi: 10.1002/cpa.3160200105.
[39]	P. H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68. doi: 10.1002/cpa.3160310103.
[40]	S. Rybicki, Periodic solutions of vibrating strings. Degree theory approach, Ann. Mat. Pura Appl. (4), 179 (2001), 197-214. doi: 10.1007/BF02505955.
[41]	M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day Inc., San Francisco, Calif., 1964.
[42]	C. Viterbo, Recent progress in periodic orbits of autonomous Hamiltonian systems and applications to symplectic geometry, in Nonlinear functional analysis (Newark, NJ, 1987), 121 of Lecture Notes in Pure and Appl. Math., 227-250. Dekker, New York, 1990.
[43]	V. Volterra, Leçons sur les Fonctions de Ligne, Gauthier-Villars, Paris, 1913.
[44]	J. von Neumann, Proposal and Analysis of a New Numerical Method for the Treatment of Hydrodynamical Shock Problems, AMP Report, (1944), 1-25.
[45]	I. R. Yukhnovskiĭ, Phase Transitions of the Second Order, World Scientific Publishing Co., Singapore, 1987. doi: 10.1142/0289.
[46]	N. J. Zabusky and M. D. Kruskal, Interaction of "solitons'' in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243. doi: 10.1103/PhysRevLett.15.240.

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