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Spectral notions of aperiodic order
Stability of nonlinear waves: Pointwise estimates
Margaret Beck, Boston University Department of Mathematics and Statistics, 111 Cummington Mall, Boston MA 02215, USA |
This is an expository article containing a brief overview of key issues related to the stability of nonlinear waves, an introduction to a particular technique in stability analysis known as pointwise estimates, and two applications of this technique: time-periodic shocks in viscous conservation laws [
References:
[1] |
M. Beck, T. Nguyen, B. Sandstede and K. Zumbrun,
Toward nonlinear stability of sources via a modified Burgers equation, Phys. D, 241 (2012), 382-392.
doi: 10.1016/j.physd.2011.10.018. |
[2] |
M. Beck, T. T. Nguyen, B. Sandstede and K. Zumbrun,
Nonlinear stability of source defects in the complex Ginzburg-Landau equation, Nonlinearity, 27 (2014), 739-786.
doi: 10.1088/0951-7715/27/4/739. |
[3] |
M. Beck, B. Sandstede and K. Zumbrun,
Nonlinear stability of time-periodic viscous shocks, Arch. Ration. Mech. Anal., 196 (2010), 1011-1076.
doi: 10.1007/s00205-009-0274-1. |
[4] |
N. Bekki and B. Nozaki,
Formations of spatial patterns and holes in the generalized Ginzburg-Landau equation, Phys. Lett. A, 110 (1985), 133-135.
doi: 10.1016/0375-9601(85)90759-5. |
[5] |
J. Bricmont, A. Kupiainen and G. Lin,
Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Communications on Pure and Applied Mathematics, 47 (1994), 893-922.
doi: 10.1002/cpa.3160470606. |
[6] |
A. Doelman,
Breaking the hidden symmetry in the ginzburg-landau equation, Physica D, 97 (1996), 398-428.
doi: 10.1016/0167-2789(95)00303-7. |
[7] |
A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains Mem. Amer. Math. Soc. 199 (2009), ⅷ+105pp.
doi: 10.1090/memo/0934. |
[8] |
K. -J. Engel and R. Nagel,
One-parameter Semigroups for Linear Evolution Equations vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. |
[9] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations Springer-Verlag, Berlin, 1981. |
[10] |
L. N. Howard and N. Kopell,
Slowly varying waves and shock structures in reaction-diffusion equations, Studies in Appl. Math., 56 (1976/77), 95-145.
|
[11] |
P. Howard and K. Zumbrun,
Stability of undercompressive shock profiles, J. Differential Equations, 225 (2006), 308-360.
doi: 10.1016/j.jde.2005.09.001. |
[12] |
M. A. Johnson, P. Noble, L. M. Rodrigues and K. Zumbrun,
Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability, Arch. Ration. Mech. Anal., 207 (2013), 693-715.
doi: 10.1007/s00205-012-0573-9. |
[13] |
T. Kapitula and J. Rubin,
Existence and stability of standing hole solutions to complex Ginzburg-Landau equations, Nonlinearity, 13 (2000), 77-112.
doi: 10.1088/0951-7715/13/1/305. |
[14] |
T. Kapitula and K. Promislow,
Spectral and Dynamical Stability of Nonlinear Waves vol. 185 of Applied Mathematical Sciences, Springer, New York, 2013, URL http://dx.doi.org/10.1007/978-1-4614-6995-7,
doi: 10.1007/978-1-4614-6995-7. |
[15] |
J. Lega,
Traveling hole solutions of the complex Ginzburg-Landau equation: A review, Phys. D, 152/153 (2001), 269-287.
doi: 10.1016/S0167-2789(01)00174-9. |
[16] |
T.-P. Liu,
Interactions of nonlinear hyperbolic waves, in Nonlinear analysis (Taipei, 1989), World Sci. Publ., Teaneck, NJ, (1991), 171-183.
|
[17] |
T.-P. Liu,
Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math., 50 (1997), 1113-1182.
doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D. |
[18] |
C. Mascia and K. Zumbrun,
Pointwise Green function bounds for shock profiles of systems with real viscosity, Arch. Ration. Mech. Anal., 169 (2003), 177-263.
doi: 10.1007/s00205-003-0258-5. |
[19] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[20] |
S. Popp, O. Stiller, I. Aranson and L. Kramer,
Hole solutions in the 1D complex Ginzburg-Landau equation, Phys. D, 84 (1995), 398-423.
doi: 10.1016/0167-2789(95)00070-K. |
[21] |
B. Sandstede and A. Scheel,
On the structure and spectra of modulated traveling waves, Math. Nachr., 232 (2001), 39-93.
doi: 10.1002/1522-2616(200112)232:1<39::AID-MANA39>3.0.CO;2-5. |
[22] |
B. Sandstede and A. Scheel,
Defects in oscillatory media: Toward a classification, SIAM Appl Dyn Sys, 3 (2004), 1-68.
doi: 10.1137/030600192. |
[23] |
B. Sandstede, A. Scheel, G. Schneider and H. Uecker,
Diffusive mixing of periodic wave trains in reaction-diffusion systems, To appear in JDE, 252 (2012), 3541-3574.
doi: 10.1016/j.jde.2011.10.014. |
[24] |
B. Sandstede and A. Scheel,
Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal., 39 (2008), 2033-2052.
doi: 10.1137/060675587. |
[25] |
G. Schneider,
Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation, Comm. Math. Phys., 178 (1996), 679-702.
doi: 10.1007/BF02108820. |
[26] |
B. Texier and K. Zumbrun,
Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves, Methods Appl. Anal., 12 (2005), 349-380.
doi: 10.4310/MAA.2005.v12.n4.a1. |
[27] |
B. Texier and K. Zumbrun,
Galloping instability of viscous shock waves, Physica D, 237 (2008), 1553-1601.
doi: 10.1016/j.physd.2008.03.008. |
[28] |
M. van Hecke,
Building blocks of spatiotemporal intermittency, Phys. Rev. Lett., 80 (1998), 1896-1899.
|
[29] |
W. van Saarloos and P. C. Hohenberg,
Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Phys. D, 56 (1992), 303-367.
doi: 10.1016/0167-2789(92)90175-M. |
[30] |
K. Zumbrun,
Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves, Quarterly of Applied Mathematics, 69 (2011), 177-202.
doi: 10.1090/S0033-569X-2011-01221-6. |
[31] |
K. Zumbrun and P. Howard,
Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871.
doi: 10.1512/iumj.1998.47.1604. |
show all references
References:
[1] |
M. Beck, T. Nguyen, B. Sandstede and K. Zumbrun,
Toward nonlinear stability of sources via a modified Burgers equation, Phys. D, 241 (2012), 382-392.
doi: 10.1016/j.physd.2011.10.018. |
[2] |
M. Beck, T. T. Nguyen, B. Sandstede and K. Zumbrun,
Nonlinear stability of source defects in the complex Ginzburg-Landau equation, Nonlinearity, 27 (2014), 739-786.
doi: 10.1088/0951-7715/27/4/739. |
[3] |
M. Beck, B. Sandstede and K. Zumbrun,
Nonlinear stability of time-periodic viscous shocks, Arch. Ration. Mech. Anal., 196 (2010), 1011-1076.
doi: 10.1007/s00205-009-0274-1. |
[4] |
N. Bekki and B. Nozaki,
Formations of spatial patterns and holes in the generalized Ginzburg-Landau equation, Phys. Lett. A, 110 (1985), 133-135.
doi: 10.1016/0375-9601(85)90759-5. |
[5] |
J. Bricmont, A. Kupiainen and G. Lin,
Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Communications on Pure and Applied Mathematics, 47 (1994), 893-922.
doi: 10.1002/cpa.3160470606. |
[6] |
A. Doelman,
Breaking the hidden symmetry in the ginzburg-landau equation, Physica D, 97 (1996), 398-428.
doi: 10.1016/0167-2789(95)00303-7. |
[7] |
A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains Mem. Amer. Math. Soc. 199 (2009), ⅷ+105pp.
doi: 10.1090/memo/0934. |
[8] |
K. -J. Engel and R. Nagel,
One-parameter Semigroups for Linear Evolution Equations vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. |
[9] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations Springer-Verlag, Berlin, 1981. |
[10] |
L. N. Howard and N. Kopell,
Slowly varying waves and shock structures in reaction-diffusion equations, Studies in Appl. Math., 56 (1976/77), 95-145.
|
[11] |
P. Howard and K. Zumbrun,
Stability of undercompressive shock profiles, J. Differential Equations, 225 (2006), 308-360.
doi: 10.1016/j.jde.2005.09.001. |
[12] |
M. A. Johnson, P. Noble, L. M. Rodrigues and K. Zumbrun,
Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability, Arch. Ration. Mech. Anal., 207 (2013), 693-715.
doi: 10.1007/s00205-012-0573-9. |
[13] |
T. Kapitula and J. Rubin,
Existence and stability of standing hole solutions to complex Ginzburg-Landau equations, Nonlinearity, 13 (2000), 77-112.
doi: 10.1088/0951-7715/13/1/305. |
[14] |
T. Kapitula and K. Promislow,
Spectral and Dynamical Stability of Nonlinear Waves vol. 185 of Applied Mathematical Sciences, Springer, New York, 2013, URL http://dx.doi.org/10.1007/978-1-4614-6995-7,
doi: 10.1007/978-1-4614-6995-7. |
[15] |
J. Lega,
Traveling hole solutions of the complex Ginzburg-Landau equation: A review, Phys. D, 152/153 (2001), 269-287.
doi: 10.1016/S0167-2789(01)00174-9. |
[16] |
T.-P. Liu,
Interactions of nonlinear hyperbolic waves, in Nonlinear analysis (Taipei, 1989), World Sci. Publ., Teaneck, NJ, (1991), 171-183.
|
[17] |
T.-P. Liu,
Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math., 50 (1997), 1113-1182.
doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D. |
[18] |
C. Mascia and K. Zumbrun,
Pointwise Green function bounds for shock profiles of systems with real viscosity, Arch. Ration. Mech. Anal., 169 (2003), 177-263.
doi: 10.1007/s00205-003-0258-5. |
[19] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[20] |
S. Popp, O. Stiller, I. Aranson and L. Kramer,
Hole solutions in the 1D complex Ginzburg-Landau equation, Phys. D, 84 (1995), 398-423.
doi: 10.1016/0167-2789(95)00070-K. |
[21] |
B. Sandstede and A. Scheel,
On the structure and spectra of modulated traveling waves, Math. Nachr., 232 (2001), 39-93.
doi: 10.1002/1522-2616(200112)232:1<39::AID-MANA39>3.0.CO;2-5. |
[22] |
B. Sandstede and A. Scheel,
Defects in oscillatory media: Toward a classification, SIAM Appl Dyn Sys, 3 (2004), 1-68.
doi: 10.1137/030600192. |
[23] |
B. Sandstede, A. Scheel, G. Schneider and H. Uecker,
Diffusive mixing of periodic wave trains in reaction-diffusion systems, To appear in JDE, 252 (2012), 3541-3574.
doi: 10.1016/j.jde.2011.10.014. |
[24] |
B. Sandstede and A. Scheel,
Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal., 39 (2008), 2033-2052.
doi: 10.1137/060675587. |
[25] |
G. Schneider,
Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation, Comm. Math. Phys., 178 (1996), 679-702.
doi: 10.1007/BF02108820. |
[26] |
B. Texier and K. Zumbrun,
Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves, Methods Appl. Anal., 12 (2005), 349-380.
doi: 10.4310/MAA.2005.v12.n4.a1. |
[27] |
B. Texier and K. Zumbrun,
Galloping instability of viscous shock waves, Physica D, 237 (2008), 1553-1601.
doi: 10.1016/j.physd.2008.03.008. |
[28] |
M. van Hecke,
Building blocks of spatiotemporal intermittency, Phys. Rev. Lett., 80 (1998), 1896-1899.
|
[29] |
W. van Saarloos and P. C. Hohenberg,
Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Phys. D, 56 (1992), 303-367.
doi: 10.1016/0167-2789(92)90175-M. |
[30] |
K. Zumbrun,
Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves, Quarterly of Applied Mathematics, 69 (2011), 177-202.
doi: 10.1090/S0033-569X-2011-01221-6. |
[31] |
K. Zumbrun and P. Howard,
Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871.
doi: 10.1512/iumj.1998.47.1604. |






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