Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Steady-state bifurcation with Euclidean symmetry


Author: Ian Melbourne
Journal: Trans. Amer. Math. Soc. 351 (1999), 1575-1603
MSC (1991): Primary 58F14, 35B32; Secondary 35K55, 35Q55
DOI: https://doi.org/10.1090/S0002-9947-99-02147-9
MathSciNet review: 1467473
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider systems of partial differential equations equivariant under the Euclidean group $\mathbf {E}(n)$ and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when $n=1$ and $n=2$ and for reaction-diffusion equations with general $n$, reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.) In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of $\mathbf {E}(n)$. The representation theory of $\mathbf {E}(n)$ is driven by the irreducible representations of $\mathbf {O}(n-1)$. For $n=1$, this constitutes a mathematical statement of the ‘universality’ of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When $n=2$, there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of $\mathbf {O}(1)$. There are infinitely many possibilities for each $n\ge 3$.


References [Enhancements On Off] (What's this?)

  • C. Baesens and R. S. MacKay, Uniformly travelling water waves from a dynamical systems viewpoint: some insights into bifurcations from Stokes’ family, J. Fluid Mech. 241 (1992), 333–347. MR 1178621, DOI https://doi.org/10.1017/S0022112092002064
  • I. Bosch Vivancos, P. Chossat, and I. Melbourne, New planforms in systems of partial differential equations with Euclidean symmetry, Arch. Rational Mech. Anal. 131 (1995), no. 3, 199–224. MR 1354695, DOI https://doi.org/10.1007/BF00382886
  • Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344
  • M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium, Rev. of Mod. Phys. 65 (1993), 851-1112.
  • Benoit Dionne and Martin Golubitsky, Planforms in two and three dimensions, Z. Angew. Math. Phys. 43 (1992), no. 1, 36–62. MR 1149370, DOI https://doi.org/10.1007/BF00944740
  • N. Dinculeanu, Integration on locally compact spaces, Noordhoff International Publishing, Leiden, 1974. Translated from the Romanian. MR 0360981
  • Martin Golubitsky and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, Applied Mathematical Sciences, vol. 51, Springer-Verlag, New York, 1985. MR 771477
  • Martin Golubitsky, Ian Stewart, and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. II, Applied Mathematical Sciences, vol. 69, Springer-Verlag, New York, 1988. MR 950168
  • M. Haragus. Reduction of PDEs on unbounded domains. Application: unsteady water waves problem, Preprint, Institut Non Linéaire de Nice, 1994.
  • Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • G. Iooss and J. Los, Bifurcation of spatially quasi-periodic solutions in hydrodynamic stability problems, Nonlinearity 3 (1990), no. 3, 851–871. MR 1067084
  • G. Iooss, A. Mielke, and Y. Demay, Theory of steady Ginzburg-Landau equation, in hydrodynamic stability problems, European J. Mech. B Fluids 8 (1989), no. 3, 229–268. MR 1008662
  • S. Ito. Unitary representations of some linear groups II, Nagoya Math. J. 5 (1953), 79-96.
  • Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
  • Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition, Dover Publications, Inc., New York, 1976. MR 0422992
  • Klaus Kirchgässner, Wave-solutions of reversible systems and applications, J. Differential Equations 45 (1982), no. 1, 113–127. MR 662490, DOI https://doi.org/10.1016/0022-0396%2882%2990058-4
  • L. D. Landau and E. M. Lifshitz, Statistical physics, Course of Theoretical Physics, Vol. 5, Pergamon Press Ltd., London-Paris; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1958. Translated from the Russian by E. Peierls and R. F. Peierls. MR 0136378
  • Serge Lang, Real and functional analysis, 3rd ed., Graduate Texts in Mathematics, vol. 142, Springer-Verlag, New York, 1993. MR 1216137
  • G. W. Mackey. Induced representations of locally compact groups I, Annals of Math. 55 (1952), 101-139.
  • I. Melbourne. Derivation of the time-dependent Ginzburg-Landau equation on the line. J. Nonlin. Sci. 8 (1998), 1–15.
  • Louis Michel, Symmetry defects and broken symmetry. Configurations. Hidden symmetry, Rev. Modern Phys. 52 (1980), no. 3, 617–651. MR 578143, DOI https://doi.org/10.1103/RevModPhys.52.617
  • Alexander Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci. 10 (1988), no. 1, 51–66. MR 929221, DOI https://doi.org/10.1002/mma.1670100105
  • Alexander Mielke, Reduction of PDEs on domains with several unbounded directions: a first step towards modulation equations, Z. Angew. Math. Phys. 43 (1992), no. 3, 449–470 (English, with German summary). MR 1166967, DOI https://doi.org/10.1007/BF00946240
  • A. C. Newell and J. A. Whitehead. Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (1969), 279-303.
  • Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
  • David Ruelle, Bifurcations in the presence of a symmetry group, Arch. Rational Mech. Anal. 51 (1973), 136–152. MR 348796, DOI https://doi.org/10.1007/BF00247751
  • D. H. Sattinger, Group representation theory, bifurcation theory and pattern formation, J. Functional Analysis 28 (1978), no. 1, 58–101. MR 493378, DOI https://doi.org/10.1016/0022-1236%2878%2990080-0
  • David H. Sattinger, Group-theoretic methods in bifurcation theory, Lecture Notes in Mathematics, vol. 762, Springer, Berlin, 1979. With an appendix entitled “How to find the symmetry group of a differential equation” by Peter Olver. MR 551626
  • H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in ${\bf R}^n$, Chinese Ann. Math. Ser. B 16 (1995), no. 4, 407–412. A Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), no. 6, 797. MR 1380578
  • L. A. Segel. Distant side-walls cause slow amplitude modulation of cellular convection, J. Fluid Mech. 38 (1969), 203-224.
  • Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • A. Vanderbauwhede, Local bifurcation and symmetry, Research Notes in Mathematics, vol. 75, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697724

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F14, 35B32, 35K55, 35Q55

Retrieve articles in all journals with MSC (1991): 58F14, 35B32, 35K55, 35Q55


Additional Information

Ian Melbourne
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3476
MR Author ID: 123300
Email: ism@math.uh.edu

Received by editor(s): May 22, 1996
Received by editor(s) in revised form: December 6, 1996
Additional Notes: Supported in part by NSF Grant DMS-9403624 and by ONR Grant N00014-94-1-0317
Article copyright: © Copyright 1999 American Mathematical Society