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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

 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

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.
Citation: Franco Cardin, Alberto Lovison. Finite mechanical proxies for a class of reducible continuum systems. Networks & 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.  Google Scholar [2] H. Amann and E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations, Houston J. Math., 7 (1981), 147-174.  Google Scholar [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.  Google Scholar [4] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), 127-166. doi: 10.1007/BF01215273.  Google Scholar [5] A. Ambrosetti, Critical points and nonlinear variational problems, Mém. Soc. Math. France (N.S.), (1992), 139 pp.  Google Scholar [6] S. S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370. doi: 10.2307/2321203.  Google Scholar [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.  Google Scholar [8] D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69-87. doi: 10.1007/s003320010010.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] M. Cappiello, Pseudodifferential parametrices of infinite order for SG-hyperbolic problems, Rend. Sem. Mat. Univ. Politec. Torino, 61 (2003), 411-441.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] M. Degiovanni, On Morse theory for continuous functionals, Conf. Semin. Mat. Univ. Bari, 290 (2003), 1-22.  Google Scholar [25] A. Di Carlo, private, communication., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [28] G. M. L. Gladwell, Inverse Problems in Vibration, 119 of Solid Mechanics and its Applications. Kluwer Academic Publishers, Dordrecht, second edition, 2004.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] A. Lovison, Generating functions and finite parameter reductions in fields theory, Bollettino Della Unione Matematica Italiana, 8A (2005), {569-572}. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] P. H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68. doi: 10.1002/cpa.3160310103.  Google Scholar [40] S. Rybicki, Periodic solutions of vibrating strings. Degree theory approach, Ann. Mat. Pura Appl. (4), 179 (2001), 197-214. doi: 10.1007/BF02505955.  Google Scholar [41] M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day Inc., San Francisco, Calif., 1964.  Google Scholar [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.  Google Scholar [43] V. Volterra, Leçons sur les Fonctions de Ligne, Gauthier-Villars, Paris, 1913. Google Scholar [44] J. von Neumann, Proposal and Analysis of a New Numerical Method for the Treatment of Hydrodynamical Shock Problems, AMP Report, (1944), 1-25. Google Scholar [45] I. R. Yukhnovskiĭ, Phase Transitions of the Second Order, World Scientific Publishing Co., Singapore, 1987. doi: 10.1142/0289.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] H. Amann and E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations, Houston J. Math., 7 (1981), 147-174.  Google Scholar [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.  Google Scholar [4] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), 127-166. doi: 10.1007/BF01215273.  Google Scholar [5] A. Ambrosetti, Critical points and nonlinear variational problems, Mém. Soc. Math. France (N.S.), (1992), 139 pp.  Google Scholar [6] S. S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370. doi: 10.2307/2321203.  Google Scholar [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.  Google Scholar [8] D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69-87. doi: 10.1007/s003320010010.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] M. Cappiello, Pseudodifferential parametrices of infinite order for SG-hyperbolic problems, Rend. Sem. Mat. Univ. Politec. Torino, 61 (2003), 411-441.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] M. Degiovanni, On Morse theory for continuous functionals, Conf. Semin. Mat. Univ. Bari, 290 (2003), 1-22.  Google Scholar [25] A. Di Carlo, private, communication., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [28] G. M. L. Gladwell, Inverse Problems in Vibration, 119 of Solid Mechanics and its Applications. Kluwer Academic Publishers, Dordrecht, second edition, 2004.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] A. Lovison, Generating functions and finite parameter reductions in fields theory, Bollettino Della Unione Matematica Italiana, 8A (2005), {569-572}. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] P. H. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68. doi: 10.1002/cpa.3160310103.  Google Scholar [40] S. Rybicki, Periodic solutions of vibrating strings. Degree theory approach, Ann. Mat. Pura Appl. (4), 179 (2001), 197-214. doi: 10.1007/BF02505955.  Google Scholar [41] M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day Inc., San Francisco, Calif., 1964.  Google Scholar [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.  Google Scholar [43] V. Volterra, Leçons sur les Fonctions de Ligne, Gauthier-Villars, Paris, 1913. Google Scholar [44] J. von Neumann, Proposal and Analysis of a New Numerical Method for the Treatment of Hydrodynamical Shock Problems, AMP Report, (1944), 1-25. Google Scholar [45] I. R. Yukhnovskiĭ, Phase Transitions of the Second Order, World Scientific Publishing Co., Singapore, 1987. doi: 10.1142/0289.  Google Scholar [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.  Google Scholar
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