Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T08:51:12.720Z Has data issue: false hasContentIssue false
  • English
  • Français

SPATIAL HAMILTONIAN IDENTITIES FOR NONLOCALLY COUPLED SYSTEMS

Part of: Representations of solutions Infinite-dimensional Hamiltonian systems Nonlinear integral equations Infinite-dimensional dissipative dynamical systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  14 November 2018

BENTE BAKKER  and
ARND SCHEEL
BENTE BAKKER
Affiliation:
Department of Mathematics, VU Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands; bente.h.bakker@gmail.com
ARND SCHEEL
Affiliation:
School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, 55455, USA; scheel@umn.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler–Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether’s theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler–Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.

MSC classification

Primary: 35S30: Fourier integral operators 45G15: Systems of nonlinear integral equations
Secondary: 35C07: Traveling wave solutions 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 37L10: Normal forms, center manifold theory, bifurcation theory 37L45: Hyperbolicity; Lyapunov functions
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e22
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

References

Abraham, R. and Marsden, J. E., Foundations of Mechanics, Vol. 36 (Benjamin/Cummings Publishing Company, Reading, MA, 1978).Google Scholar
Anderson, T., Faye, G., Scheel, A. and Stauffer, D., ‘Pinning and unpinning in nonlocal systems’, J. Dynam. Differential Equations 28(3–4) (2016), 897–923.Google Scholar
Angenent, S. and Vorst, R. v. d., ‘A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology’, Math. Z. 231(2) (1999), 203–248.Google Scholar
Bakker, B., Berg, J. B. v. d. and Vandervorst, R., ‘A Floer homology approach to travelling waves in reaction–diffusion equations on cylinders’, SIAM J. Appl. Dyn. Syst. (in press).Google Scholar
Bates, P. W., ‘On some nonlocal evolution equations arising in materials science’, inNonlinear Dynamics and Evolution Equations, Fields Institute Communications, 48 (American Mathematical Society, Providence, RI, 2006), 13–52.Google Scholar
Bates, P. W. and Chmaj, A., ‘An integrodifferential model for phase transitions: stationary solutions in higher space dimensions’, J. Stat. Phys. 95(5) (1999), 1119–1139.Google Scholar
Bates, P. W., Fife, P. C., Ren, X. and Wang, X., ‘Traveling waves in a convolution model for phase transitions’, Arch. Ration. Mech. Anal. 138(2) (1997), 105–136.Google Scholar
Bridges, T. J., ‘Multi-symplectic structures and wave propagation’, Math. Proc. Cambridge Philos. Soc. 121(1) (1997), 147–190.Google Scholar
Bridges, T. J. and Laine-Pearson, F. E., ‘Multisymplectic relative equilibria, multiphase wavetrains, and coupled NLS equations’, Stud. Appl. Math. 107(2) (2001), 137–155.Google Scholar
Bridges, T. J. and Laine-Pearson, F. E., ‘Nonlinear counterpropagating waves, multisymplectic geometry, and the instability of standing waves’, SIAM J. Appl. Math. 64(6) (2004), 2096–2120.Google Scholar
Cabré, X. and Sire, Y., ‘Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates’, Ann. Inst. H. Poincaré Anal. Non Linéaire 31(1) (2014), 23–53. ISSN 0294-1449.Google Scholar
Cabré, X. and Solà-Morales, J., ‘Layer solutions in a half-space for boundary reactions’, Comm. Pure Appl. Math. 58(12) (2005), 1678–1732. ISSN 0010-3640. URL https://doi.org/10.1002/cpa.20093.Google Scholar
Du, Q., Gunzburger, M., Lehoucq, R. B. and Zhou, K., ‘Analysis and approximation of nonlocal diffusion problems with volume constraints’, SIAM Rev. 54(4) (2012), 667–696.Google Scholar
Du, Q., Gunzburger, M., Lehoucq, R. and Zhou, K., ‘Analysis of the volume-constrained peridynamic navier equation of linear elasticity’, J. Elasticity 113(2) (2013), 193–217.Google Scholar
Ehrnström, M., Groves, M. D. and Wahlén, E., ‘On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type’, Nonlinearity 25(10) (2012), 2903.Google Scholar
Ehrnström, M. and Kalisch, H. et al. , ‘Traveling waves for the Whitham equation’, Differ. Integral Equ. 22(11/12) (2009), 1193–1210.Google Scholar
Ermentrout, G. B. and McLeod, J. B., ‘Existence and uniqueness of travelling waves for a neural network’, Proc. Roy. Soc. Edinburgh Sect. A 123(3) (1993), 461–478.Google Scholar
Faye, G. and Scheel, A., ‘Fredholm properties of nonlocal differential operators via spectral flow’, Indiana Univ. Math. J. 63(5) (2014), 1311–1348. ISSN 0022-2518. URL https://doi.org/10.1512/iumj.2014.63.5383.Google Scholar
Faye, G. and Scheel, A., ‘Existence of pulses in excitable media with nonlocal coupling’, Adv. Math. 270 (2015), 400–456.Google Scholar
Faye, G. and Scheel, A., ‘Center manifolds without a phase space’, Trans. Amer. Math. Soc. 370(8) (2018), 5843–5885. ISSN 0002-9947. URL https://doi.org/10.1090/tran/7190.Google Scholar
Frank, R. L., Lenzmann, E. and Silvestre, L., ‘Uniqueness of radial solutions for the fractional Laplacian’, Comm. Pure Appl. Math. 69(9) (2016), 1671–1726. ISSN 1097-0312.Google Scholar
Gui, C., ‘Hamiltonian identities for elliptic partial differential equations’, J. Funct. Anal. 254(4) (2008), 904–933.Google Scholar
Gunzburger, M. and Lehoucq, R. B., ‘A nonlocal vector calculus with application to nonlocal boundary value problems’, Multiscale Model. Simul. 8(5) (2010), 1581–1598.Google Scholar
Härterich, J., Sandstede, B. and Scheel, A., ‘Exponential dichotomies for linear non-autonomous functional differential equations of mixed type’, Indiana Univ. Math. J. 51(5) (2002), 1081–1109. ISSN 0022-2518.Google Scholar
Hupkes, H. J. and Sandstede, B., ‘Traveling pulse solutions for the discrete FitzHugh–Nagumo system’, SIAM J. Appl. Dyn. Syst. 9(3) (2010), 827–882. ISSN 1536-0040.Google Scholar
Hupkes, H. J. and Verduyn Lunel, S. M., ‘Center manifold theory for functional differential equations of mixed type’, J. Dynam. Differential Equations 19(2) (2007), 497–560. ISSN 1040-7294.Google Scholar
Iooss, G. and Kirchgässner, K., ‘Travelling waves in a chain of coupled nonlinear oscillators’, Comm. Math. Phys. 211(2) (2000), 439–464. ISSN 0010-3616. URL https://doi.org/10.1007/s002200050821.Google Scholar
Kirchgässner, K., ‘Wave-solutions of reversible systems and applications’, J. Differ. Equ. 45(1) (1982), 113–127. ISSN 0022-0396.Google Scholar
Kuksin, S. B., Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19 (Oxford University Press, Oxford, 2000).Google Scholar
Lee, J. M., Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218 (Springer, New York, 2003).Google Scholar
Lloyd, D. J. B. and Scheel, A., ‘Continuation and bifurcation of grain boundaries in the Swift–Hohenberg equation’, SIAM J. Appl. Dyn. Syst. 16(1) (2017), 252–293.Google Scholar
Mallet-Paret, J. and Verduyn Lune, S., ‘Mixed-type functional differential equations, holomorphic factorization, and applications’, inEQUADIFF 2003 (World Scientific Publishing, Hackensack, NJ, 2005), 73–89.Google Scholar
Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry, 2nd edn, Texts in Applied Mathematics, 17 (Springer, New York, 1999).Google Scholar
Mielke, A., Hamiltonian and Lagrangian Flows on Center Manifolds, Lecture Notes in Mathematics, 1489 (Springer, Berlin, 1991), ISBN 3-540-54710-X. With applications to elliptic variational problems.Google Scholar
Mielke, A., A Spatial Center Manifold Approach to Steady State Bifurcations from Spatially Periodic Patterns, Pitman Research Notes in Mathematics Series, 352 (Longman, Harlow, 1996), Ch. 4, 209–262.Google Scholar
Mischaikow, K. and Mrozek, M., ‘Conley index’, inHandbook of Dynamical Systems, Vol. 2 (North-Holland, Amsterdam, 2002), 393–460. URL https://doi.org/10.1016/S1874-575X(02)80030-3.Google Scholar
Monteiro, R. and Scheel, A., ‘Phase separation patterns from directional quenching’, J. Nonlinear Sci. 27(5) (2017), 1339–1378. ISSN 1432-1467.Google Scholar
Naumkin, P. I. and Shishmarëv, I. A., Nonlinear Nonlocal Equations in the Theory of Waves, Translations of Mathematical Monographs, 133 (American Mathematical Society, Providence, RI, 1994).Google Scholar
Pinto, D. J. and Ermentrout, G. B., ‘Spatially structured activity in synaptically coupled neuronal networks. I. Traveling fronts and pulses’, SIAM J. Appl. Math. 62(1) (2001), 206–225.Google Scholar
Scheel, A. and Tao, T., ‘Bifurcation to coherent structures in nonlocally coupled systems’, J. Dynam. Differential Equations (2017), (in press).Google Scholar
Scheel, A. and Weinburd, J., ‘Wavenumber selection via spatial parameter jump’, Philos. Trans. R. Soc. Lond. A 376(2117) (2018), 20170191, 20. ISSN 1364-503X.Google Scholar
Silling, S. A., ‘Reformulation of elasticity theory for discontinuities and long-range forces’, J. Mech. Phys. Solids 48(1) (2000), 175–209.Google Scholar
Vanderbauwhede, A. and Fiedler, B., ‘Homoclinic period blow-up in reversible and conservative systems’, Z. Angew. Math. Phys. 43(2) (1992), 292–318.Google Scholar
Whitham, G. B., Linear and Nonlinear Waves, Pure and Applied Mathematics (John Wiley & Sons, Inc., New York, 1999).Google Scholar
Wilson, H. R. and Cowan, J. D., ‘Excitatory and inhibitory interactions in localized populations of model neurons’, Biophys. J. 12(1) (1972), 1–24.Google Scholar
Wilson, H. R. and Cowan, J. D., ‘A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue’, Biol. Cybernet. 13(2) (1973), 55–80.Google Scholar