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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

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 & 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.   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.   Google Scholar

[3]

H. Amann, Multiple positive fixed points of asymptotically linear maps,, J. Functional Analysis, 17 (1974), 174.  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.  doi: 10.1007/BF01215273.  Google Scholar

[5]

A. Ambrosetti, Critical points and nonlinear variational problems,, Mém. Soc. Math. France (N.S.), (1992).   Google Scholar

[6]

S. S. Antman, The equations for large vibrations of strings,, Amer. Math. Monthly, 87 (1980), 359.  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.  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.  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.   Google Scholar

[10]

M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities,, Comm. Math. Phys., 243 (2003), 315.  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.  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.  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.   Google Scholar

[14]

F. Cardin, Global finite generating functions for field theory,, in Classical and quantum integrability (Warsaw, 59 (2001), 133.  doi: 10.4064/bc59-0-6.  Google Scholar

[15]

F. Cardin and A. Lovison, Microscopic structures from reduction of continuum nonlinear problems,, AAPP - Physical, 91 (2013), 1.  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.  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.  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.  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.  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.  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.  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, 6 (1998), 214.  doi: 10.1142/9789812792099_0013.  Google Scholar

[24]

M. Degiovanni, On Morse theory for continuous functionals,, Conf. Semin. Mat. Univ. Bari, 290 (2003), 1.   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.  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.   Google Scholar

[28]

G. M. L. Gladwell, Inverse Problems in Vibration,, 119 of Solid Mechanics and its Applications. Kluwer Academic Publishers, (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.  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.  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.   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.  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.   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).  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.  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.  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.  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.  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.  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.  doi: 10.1007/BF02505955.  Google Scholar

[41]

M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators,, Holden-Day Inc., (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, 121 (1987), 227.   Google Scholar

[43]

V. Volterra, Leçons sur les Fonctions de Ligne,, Gauthier-Villars, (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.   Google Scholar

[45]

I. R. Yukhnovskiĭ, Phase Transitions of the Second Order,, World Scientific Publishing Co., (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.  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.   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.   Google Scholar

[3]

H. Amann, Multiple positive fixed points of asymptotically linear maps,, J. Functional Analysis, 17 (1974), 174.  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.  doi: 10.1007/BF01215273.  Google Scholar

[5]

A. Ambrosetti, Critical points and nonlinear variational problems,, Mém. Soc. Math. France (N.S.), (1992).   Google Scholar

[6]

S. S. Antman, The equations for large vibrations of strings,, Amer. Math. Monthly, 87 (1980), 359.  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.  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.  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.   Google Scholar

[10]

M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities,, Comm. Math. Phys., 243 (2003), 315.  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.  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.  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.   Google Scholar

[14]

F. Cardin, Global finite generating functions for field theory,, in Classical and quantum integrability (Warsaw, 59 (2001), 133.  doi: 10.4064/bc59-0-6.  Google Scholar

[15]

F. Cardin and A. Lovison, Microscopic structures from reduction of continuum nonlinear problems,, AAPP - Physical, 91 (2013), 1.  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.  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.  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.  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.  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.  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.  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, 6 (1998), 214.  doi: 10.1142/9789812792099_0013.  Google Scholar

[24]

M. Degiovanni, On Morse theory for continuous functionals,, Conf. Semin. Mat. Univ. Bari, 290 (2003), 1.   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.  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.   Google Scholar

[28]

G. M. L. Gladwell, Inverse Problems in Vibration,, 119 of Solid Mechanics and its Applications. Kluwer Academic Publishers, (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.  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.  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.   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.  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.   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).  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.  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.  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.  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.  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.  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.  doi: 10.1007/BF02505955.  Google Scholar

[41]

M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators,, Holden-Day Inc., (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, 121 (1987), 227.   Google Scholar

[43]

V. Volterra, Leçons sur les Fonctions de Ligne,, Gauthier-Villars, (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.   Google Scholar

[45]

I. R. Yukhnovskiĭ, Phase Transitions of the Second Order,, World Scientific Publishing Co., (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.  doi: 10.1103/PhysRevLett.15.240.  Google Scholar

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