doi: 10.3934/dcdss.2020061

Bifurcation revisited along footprints of Jürgen Scheurle

Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D50931 Köln, Germany

Received  September 2017 Revised  June 2018 Published  April 2019

Actual research concerning, in particular, the occurrence of "gap-solitons" bifurcating from the continuous spectrum confirms that this part of Bifurcation Theory that started around 40 years ago flourishes. In this lecture we review the origins of "Bifurcation from the continuous spectrum" with regard to the achievements of Jürgen Scheurle and sketch how the early results dealing with the bifurcation of singular solutions have prepared the ground for present and further developments.

Citation: Tassilo Küpper. Bifurcation revisited along footprints of Jürgen Scheurle. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020061
References:
[1]

S. Alama and Y. Li, Existence of solutions for semilinear elliptic equations with indefinite linear part, J. Diff. Equat., 96 (1992), 89-115. doi: 10.1016/0022-0396(92)90145-D.

[2]

N. Bazley and T. Küpper, Branches of solutions in nonlinear eigenvalue problems, in Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchgässner), Pitman, (1981), 24-40.

[3]

V. Benci and D. Fortunato, Does bifurcation from the continuous spectrum occur?, Comm. Partial. Diff. Equat., 6 (1981), 249-272. doi: 10.1080/03605308108820176.

[4]

A. BongersH. Heinz and T. Küpper, Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains, J. Diff. Equat., 47 (1983), 327-357. doi: 10.1016/0022-0396(83)90040-2.

[5]

R. Chiapinelli and C. Stuart, Bifurcation when the linearized problem has no eigenvalues, J. Diff. Equat., 30 (1978), 296-307. doi: 10.1016/0022-0396(78)90002-5.

[6]

H. Chiba, A spectral theory of linear operators on rigged Hilbert spaces under analytical conditions, Adv. In Math., 273 (2015), 324-379. doi: 10.1016/j.aim.2015.01.001.

[7]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, Ergo. Theo. Dyn. Syst., 35 (2015), 762-834. doi: 10.1017/etds.2013.68.

[8]

Y. Demay, Bifurcation d'' un solution pour une equation de la physique de plasma, C.R. Acad. Sci. Paris, 285 (1977), 769-772.

[9]

T. DohnalD. Pelinovsky and G. Schneider, Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, J. Nonl. Sci., 19 (2009), 95-131. doi: 10.1007/s00332-008-9027-9.

[10]

T. Dohnal and H. Uecker, Bifurcation of nonlinear Bloch waves from the spectrum in the Gross-Pitaevskii equation, J. Nonl. Sci., 26 (2016), 581-618. doi: 10.1007/s00332-015-9281-6.

[11]

B. Fiedler, Egodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer-Verlag, Berlin 2001. doi: 10.1007/978-3-642-56589-2.

[12]

M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory Vol.I, Springer-Verlag, 1985. doi: 10.1007/978-1-4612-5034-0.

[13]

M. Golubitsky, D. Schaeffer and I. Stewart, Singularities and Groups in Bifurcation Theory Vol.II, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-4574-2.

[14]

H. Heinz, Über das Verzweigungsverhalten eines nichtlinearen Eigenwertproblems in der Nhe eines unendlich vielfachen Eigenwerts der Linearisierung, Manuscripta Math., 19 (1976), 105-132. doi: 10.1007/BF01275416.

[15]

H. Heinz, Existence and gap-bifurcation of multiple solutions to certain nonlinear eigenvalue problem, Nonlin. Anal., 21 (1993), 457-484. doi: 10.1016/0362-546X(93)90128-F.

[16]

H. HeinzT. Küpper and C. Stuart, Existence and bifurcation for nonlinear perturbations of the periodic Schrödinger equation, J. Diff. Equat., 100 (1992), 341-354. doi: 10.1016/0022-0396(92)90118-7.

[17]

C. Jones and T. Küpper, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17 (1986), 803-835. doi: 10.1137/0517059.

[18]

C. Jones and T. Küpper, Characterisation of bifurcation from the continuous spectrum by nodal properties, J. Diff. Equat., 54 (1984), 196-220. doi: 10.1016/0022-0396(84)90158-X.

[19]

K. Kirchgässner and J. Scheurle, Verzweigung und Stabilitaet von Loesungen semilinearer elliptischer Randwertprobleme, Jahresberichte Deutsche Mathematiker Vereinigung, 77 (1975), 39-54.

[20]

K. Kirchgässner and J. Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Diff. Equat., 32 (1979), 119-148. doi: 10.1016/0022-0396(79)90055-X.

[21]

K. Kirchgässner and J. Scheurle, Bifurcation from the continuous spectrum and singular solutions, in Trends in Applications of pure Mathematics to Mechanics, Vol. III (ed. R.Knops), Pitman, London, 11 (1981), 138-154.

[22]

K. Kirchgässner and J. Scheurle, Global branches of periodic solutions of reversible systems, in Contributions to Nonlinear Partial Differential Equations (eds. H.Brezis), Pitman, London, 50 (1981), 103-130.

[23]

M. KunzeT. KüpperV. Shapiro and S. Turitsyn, Nonlinearity solitary waves with Gaussian tails, Physica D, 128 (1999), 273-295. doi: 10.1016/S0167-2789(98)00297-8.

[24]

T. Küpper, On minimal nonlinearities which permit bifurcation from the continuous spectrum, Math. Meth. Appl. Sci., 1 (1979), 572-580. doi: 10.1002/mma.1670010414.

[25]

T. Küpper, Singuläre Verzweigungsprobleme, Habilitationsschrift, Universitaet zu Köln, 1979.

[26]

T. Küpper, The lowest point of the continuous spectrum as bifurcation point, J. Diff. Equat., 34 (1979), 212-217. doi: 10.1016/0022-0396(79)90005-6.

[27]

T. Küpper, Verzweigung aus dem wesentlichen Spektrum, GAMM-Mitteilungen, 1 (1991), 11-22.

[28]

T. Küpper, Nonlinear Phenomena: Final Report on Joint Research Program with Chinese Universities, Köln., 2005.

[29]

T. Küpper and T. Mrziglod, On the bifurcation structure of nonlinear perturbations of Hill`s equations at boundary points of the continuous sprectrum, SIAM J. Math. Anal., 26 (1995), 1284-1305. doi: 10.1137/S0036141093250876.

[30]

T. Küpper and D. Riemer, Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonl. Anal (TMA), 3 (1979), 555-561. doi: 10.1016/0362-546X(79)90073-7.

[31]

T. Küpper and C. Stuart, Bifurcation into gaps in the essential spectrum, J. Reine Angew. Math., 409 (1990), 1-34. doi: 10.1007/978-94-009-0659-4_32.

[32]

T. Küpper and C. Stuart, Gap-bifurcation for nonlinear perturbations of Hill's equation, J. Reine Angew. Math., 410 (1990), 23-52.

[33]

T. Küpper and C. Stuart, Bifurcation at boundary points of the continuous spectrum, in Progress in Nonlinear Differential Equations and their Applications (eds. N.G. Lloyd, W.M. Ni, L.A. Peletier and J. Serrin), 7 (1992), 287-297.

[34]

T. Küpper and C. Stuart, Necessary and sufficient conditions for gap-bifurcation, Nonlin. Anal., 18 (1992), 893-903. doi: 10.1016/0362-546X(92)90230-C.

[35]

T. Mrziglod, Untersuchungen zur Loesungsstruktur nichtlinearer Hill'scher Gleichungen, Habilitationsschrift Univ.- Köln, 1996.

[36]

D. Pelinovsky, A. A. Sukhorukov and Y. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials, Physical Review E, 70 (2004), 036618, 17pp. doi: 10.1103/PhysRevE.70.036618.

[37]

A. Plate, Verzweigung aus dem wesentlichen Spektrum, Dissertation Universitt Hannover, 1992.

[38]

J. Scheurle, Newton iterations without inverting the derivative, Math. Meth. Appl. Sci., 1 (1979), 514-529. doi: 10.1002/mma.1670010409.

[39]

J. Scheurle, Ein selektives Iterationsverfahren und Verzweigungsprobleme, Dissertation, Universitt Stuttgart, 1975.

[40]

J. Scheurle, A selective iteration procedure for Taylor's stability problem, Proc. of the 2nd GAMM-conference on Numerical Methods in Fluid Mech., (1977), 176-183.

[41]

J. Scheurle, Ein selektives Projektions-Iterationsverfahren und Anwendungen auf Verzweigungsprobleme, Num. Math., 29 (1977), 11-35. doi: 10.1007/BF01389310.

[42]

J. Scheurle, Selective iteration and applications, J. Math.Anal. Appl., 59 (1977), 596-616. doi: 10.1016/0022-247X(77)90084-1.

[43]

J. Scheurle, Bifurcation of a stationary solution of a dynamical system into n-dimensional tori of quasiperiodic, in Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Math., 730 (1979), 442-454.

[44]

J. Scheurle, Über die Konstruktion invarianter Tori, welche von einer stationaeren Grundloesung eines reversiblen dynamischen Systems abzweigen, in Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations (eds. J. Albrecht, L. Collatz and K. Kirchgässner)

[45]

J. Scheurle, Verzweigung Quasiperiodischer L Sungen bei Reversiblen Dynamischen Systemen, Habilitationsschrift Universität Stuttgart, 1980.

[46]

J. Scheurle, Quasiperiodic solutions of a semilinear equation in a two-dimensional strip, in Dynamical Problems in Mathematical Physics (eds. B. Brosowski and E. Martensen), 26, Lang-Verlag, (1983), 201-223.

[47]

J. Scheurle and K. Kirchgässner, Bifurcation of non-periodic solutions of some semi-linear equations in unbounded domains, in Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchg ssner), London: Pitman, (1981), 41-59.

[48]

C. A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. Ser., 45 (1982), 169-192. doi: 10.1112/plms/s3-45.1.169.

[49]

C. A. Stuart, Bifurcation for Neumann problems without eigenvalues, J. Diff. Equat., 36 (1980), 391-407. doi: 10.1016/0022-0396(80)90057-1.

[50]

C. A. Stuart, Bifurcation when the linearisation has no eigenvalues, J. Funct. Anal., 38 (1980), 169-187. doi: 10.1016/0022-1236(80)90063-4.

[51]

C. A. Stuart, Bifurcation from the continuous spectrum in LP(R), in Bifurcation: Analysis, Algorithms, Applications ISNM (eds. T. Küpper, R. Seydel and H. Troger), Birkhäuser, 79 (1987), 306-318.

[52]

C. A. Stuart, Bifurcation of homoclinic orbits and bifurcation from the essential spectrum, SIAM J. Math. Anal., 20 (1989), 1145-1171. doi: 10.1137/0520076.

[53]

C. A. Stuart, Bifurcation into Spectral Gaps, Bulletin of the Belgian Mathematical Society, 1995.

show all references

References:
[1]

S. Alama and Y. Li, Existence of solutions for semilinear elliptic equations with indefinite linear part, J. Diff. Equat., 96 (1992), 89-115. doi: 10.1016/0022-0396(92)90145-D.

[2]

N. Bazley and T. Küpper, Branches of solutions in nonlinear eigenvalue problems, in Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchgässner), Pitman, (1981), 24-40.

[3]

V. Benci and D. Fortunato, Does bifurcation from the continuous spectrum occur?, Comm. Partial. Diff. Equat., 6 (1981), 249-272. doi: 10.1080/03605308108820176.

[4]

A. BongersH. Heinz and T. Küpper, Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains, J. Diff. Equat., 47 (1983), 327-357. doi: 10.1016/0022-0396(83)90040-2.

[5]

R. Chiapinelli and C. Stuart, Bifurcation when the linearized problem has no eigenvalues, J. Diff. Equat., 30 (1978), 296-307. doi: 10.1016/0022-0396(78)90002-5.

[6]

H. Chiba, A spectral theory of linear operators on rigged Hilbert spaces under analytical conditions, Adv. In Math., 273 (2015), 324-379. doi: 10.1016/j.aim.2015.01.001.

[7]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, Ergo. Theo. Dyn. Syst., 35 (2015), 762-834. doi: 10.1017/etds.2013.68.

[8]

Y. Demay, Bifurcation d'' un solution pour une equation de la physique de plasma, C.R. Acad. Sci. Paris, 285 (1977), 769-772.

[9]

T. DohnalD. Pelinovsky and G. Schneider, Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, J. Nonl. Sci., 19 (2009), 95-131. doi: 10.1007/s00332-008-9027-9.

[10]

T. Dohnal and H. Uecker, Bifurcation of nonlinear Bloch waves from the spectrum in the Gross-Pitaevskii equation, J. Nonl. Sci., 26 (2016), 581-618. doi: 10.1007/s00332-015-9281-6.

[11]

B. Fiedler, Egodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer-Verlag, Berlin 2001. doi: 10.1007/978-3-642-56589-2.

[12]

M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory Vol.I, Springer-Verlag, 1985. doi: 10.1007/978-1-4612-5034-0.

[13]

M. Golubitsky, D. Schaeffer and I. Stewart, Singularities and Groups in Bifurcation Theory Vol.II, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-4574-2.

[14]

H. Heinz, Über das Verzweigungsverhalten eines nichtlinearen Eigenwertproblems in der Nhe eines unendlich vielfachen Eigenwerts der Linearisierung, Manuscripta Math., 19 (1976), 105-132. doi: 10.1007/BF01275416.

[15]

H. Heinz, Existence and gap-bifurcation of multiple solutions to certain nonlinear eigenvalue problem, Nonlin. Anal., 21 (1993), 457-484. doi: 10.1016/0362-546X(93)90128-F.

[16]

H. HeinzT. Küpper and C. Stuart, Existence and bifurcation for nonlinear perturbations of the periodic Schrödinger equation, J. Diff. Equat., 100 (1992), 341-354. doi: 10.1016/0022-0396(92)90118-7.

[17]

C. Jones and T. Küpper, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17 (1986), 803-835. doi: 10.1137/0517059.

[18]

C. Jones and T. Küpper, Characterisation of bifurcation from the continuous spectrum by nodal properties, J. Diff. Equat., 54 (1984), 196-220. doi: 10.1016/0022-0396(84)90158-X.

[19]

K. Kirchgässner and J. Scheurle, Verzweigung und Stabilitaet von Loesungen semilinearer elliptischer Randwertprobleme, Jahresberichte Deutsche Mathematiker Vereinigung, 77 (1975), 39-54.

[20]

K. Kirchgässner and J. Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Diff. Equat., 32 (1979), 119-148. doi: 10.1016/0022-0396(79)90055-X.

[21]

K. Kirchgässner and J. Scheurle, Bifurcation from the continuous spectrum and singular solutions, in Trends in Applications of pure Mathematics to Mechanics, Vol. III (ed. R.Knops), Pitman, London, 11 (1981), 138-154.

[22]

K. Kirchgässner and J. Scheurle, Global branches of periodic solutions of reversible systems, in Contributions to Nonlinear Partial Differential Equations (eds. H.Brezis), Pitman, London, 50 (1981), 103-130.

[23]

M. KunzeT. KüpperV. Shapiro and S. Turitsyn, Nonlinearity solitary waves with Gaussian tails, Physica D, 128 (1999), 273-295. doi: 10.1016/S0167-2789(98)00297-8.

[24]

T. Küpper, On minimal nonlinearities which permit bifurcation from the continuous spectrum, Math. Meth. Appl. Sci., 1 (1979), 572-580. doi: 10.1002/mma.1670010414.

[25]

T. Küpper, Singuläre Verzweigungsprobleme, Habilitationsschrift, Universitaet zu Köln, 1979.

[26]

T. Küpper, The lowest point of the continuous spectrum as bifurcation point, J. Diff. Equat., 34 (1979), 212-217. doi: 10.1016/0022-0396(79)90005-6.

[27]

T. Küpper, Verzweigung aus dem wesentlichen Spektrum, GAMM-Mitteilungen, 1 (1991), 11-22.

[28]

T. Küpper, Nonlinear Phenomena: Final Report on Joint Research Program with Chinese Universities, Köln., 2005.

[29]

T. Küpper and T. Mrziglod, On the bifurcation structure of nonlinear perturbations of Hill`s equations at boundary points of the continuous sprectrum, SIAM J. Math. Anal., 26 (1995), 1284-1305. doi: 10.1137/S0036141093250876.

[30]

T. Küpper and D. Riemer, Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonl. Anal (TMA), 3 (1979), 555-561. doi: 10.1016/0362-546X(79)90073-7.

[31]

T. Küpper and C. Stuart, Bifurcation into gaps in the essential spectrum, J. Reine Angew. Math., 409 (1990), 1-34. doi: 10.1007/978-94-009-0659-4_32.

[32]

T. Küpper and C. Stuart, Gap-bifurcation for nonlinear perturbations of Hill's equation, J. Reine Angew. Math., 410 (1990), 23-52.

[33]

T. Küpper and C. Stuart, Bifurcation at boundary points of the continuous spectrum, in Progress in Nonlinear Differential Equations and their Applications (eds. N.G. Lloyd, W.M. Ni, L.A. Peletier and J. Serrin), 7 (1992), 287-297.

[34]

T. Küpper and C. Stuart, Necessary and sufficient conditions for gap-bifurcation, Nonlin. Anal., 18 (1992), 893-903. doi: 10.1016/0362-546X(92)90230-C.

[35]

T. Mrziglod, Untersuchungen zur Loesungsstruktur nichtlinearer Hill'scher Gleichungen, Habilitationsschrift Univ.- Köln, 1996.

[36]

D. Pelinovsky, A. A. Sukhorukov and Y. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials, Physical Review E, 70 (2004), 036618, 17pp. doi: 10.1103/PhysRevE.70.036618.

[37]

A. Plate, Verzweigung aus dem wesentlichen Spektrum, Dissertation Universitt Hannover, 1992.

[38]

J. Scheurle, Newton iterations without inverting the derivative, Math. Meth. Appl. Sci., 1 (1979), 514-529. doi: 10.1002/mma.1670010409.

[39]

J. Scheurle, Ein selektives Iterationsverfahren und Verzweigungsprobleme, Dissertation, Universitt Stuttgart, 1975.

[40]

J. Scheurle, A selective iteration procedure for Taylor's stability problem, Proc. of the 2nd GAMM-conference on Numerical Methods in Fluid Mech., (1977), 176-183.

[41]

J. Scheurle, Ein selektives Projektions-Iterationsverfahren und Anwendungen auf Verzweigungsprobleme, Num. Math., 29 (1977), 11-35. doi: 10.1007/BF01389310.

[42]

J. Scheurle, Selective iteration and applications, J. Math.Anal. Appl., 59 (1977), 596-616. doi: 10.1016/0022-247X(77)90084-1.

[43]

J. Scheurle, Bifurcation of a stationary solution of a dynamical system into n-dimensional tori of quasiperiodic, in Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Math., 730 (1979), 442-454.

[44]

J. Scheurle, Über die Konstruktion invarianter Tori, welche von einer stationaeren Grundloesung eines reversiblen dynamischen Systems abzweigen, in Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations (eds. J. Albrecht, L. Collatz and K. Kirchgässner)

[45]

J. Scheurle, Verzweigung Quasiperiodischer L Sungen bei Reversiblen Dynamischen Systemen, Habilitationsschrift Universität Stuttgart, 1980.

[46]

J. Scheurle, Quasiperiodic solutions of a semilinear equation in a two-dimensional strip, in Dynamical Problems in Mathematical Physics (eds. B. Brosowski and E. Martensen), 26, Lang-Verlag, (1983), 201-223.

[47]

J. Scheurle and K. Kirchgässner, Bifurcation of non-periodic solutions of some semi-linear equations in unbounded domains, in Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchg ssner), London: Pitman, (1981), 41-59.

[48]

C. A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. Ser., 45 (1982), 169-192. doi: 10.1112/plms/s3-45.1.169.

[49]

C. A. Stuart, Bifurcation for Neumann problems without eigenvalues, J. Diff. Equat., 36 (1980), 391-407. doi: 10.1016/0022-0396(80)90057-1.

[50]

C. A. Stuart, Bifurcation when the linearisation has no eigenvalues, J. Funct. Anal., 38 (1980), 169-187. doi: 10.1016/0022-1236(80)90063-4.

[51]

C. A. Stuart, Bifurcation from the continuous spectrum in LP(R), in Bifurcation: Analysis, Algorithms, Applications ISNM (eds. T. Küpper, R. Seydel and H. Troger), Birkhäuser, 79 (1987), 306-318.

[52]

C. A. Stuart, Bifurcation of homoclinic orbits and bifurcation from the essential spectrum, SIAM J. Math. Anal., 20 (1989), 1145-1171. doi: 10.1137/0520076.

[53]

C. A. Stuart, Bifurcation into Spectral Gaps, Bulletin of the Belgian Mathematical Society, 1995.

Figure 1.  Phase portrait for $ \lambda >0 $
Figure 2.  Heteroclinic orbit (singular solution)
Figure 3.  (a) λ > 0, (b) λ < 0
Figure 4.  Prof. Tassilo Küpper and Prof. Jürgen Scheurle
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