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April  2017, 10(2): 191-211. doi: 10.3934/dcdss.2017010

Stability of nonlinear waves: Pointwise estimates

Margaret Beck

Margaret Beck, Boston University Department of Mathematics and Statistics, 111 Cummington Mall, Boston MA 02215, USA

Received  July 2015 Revised  November 2016 Published  January 2017

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 [3] and source defects in reaction diffusion equations [1, 2].

Keywords: Green's function, pointwise estimates, stability, coherent structure.
Mathematics Subject Classification: 35Kxx, 34Dxx.
Citation: Margaret Beck. Stability of nonlinear waves: Pointwise estimates. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 191-211. doi: 10.3934/dcdss.2017010
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.

Figure 1.  Typical spectra of linear operators that are spectrally stable in a strong sense: $\sup \mathrm{Re} \sigma(\mathcal{L}) < 0$. On the left we see a half line of essential spectrum and an isolated eigenvalue (the cross), and on the right we see a parabolic region of essential spectrum and an isolated eigenvalue.
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Figure 2.  Typical spectra of linear operators that are spectrally stable in a weaker sense: $\sup \mathrm{Re} \sigma(\mathcal{L}) = 0$. On the left we see a half line of essential spectrum and an isolated eigenvalue (the cross) on the imaginary axis, and on the right we see a parabolic region of essential spectrum touching the imaginary axis and an embedded eigenvalue (denoted now in red for visual clarity) at the origin.
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Figure 3.  Floquet spectrum of a spectrally stable viscous shock near the origin. Note the spectrum is non-unique, as it can be shifted by any integer multiple of $2\pi{\rm{i}}$, and hence the parabolas repeat infinitely many times up and down the imaginary axis. There are two embedded eigenvalues at the origin, due to translations in space and time.
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Figure 4.  Left panel: original vertical contour with real part $\mu$ used in 3.4. Right panel: deformed contour used to obtain bounds on the Green's function. The parameters $\epsilon$ and $r$ can be chosen to be small and to optimize the resultant bounds.
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Figure 5.  On the left is a diagram of the profile of a source as a function of $x$ for a fixed value of $t$, with the motion of perturbations, relative to the speed of the defect core, indicated by the red arrows and the group velocities of the asymptotic wave trains. The right panel shows the behavior of small phase $\phi$ or wave number $\phi_x$ perturbation of a wave train: to leading order, they are transported with speed given by the group velocity $c_g$ without changing their shape [7].
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Figure 6.  On the left is a sketch of the space-time diagram of a perturbed source. The defect core will adjust in response to an imposed perturbation (although this is not depicted), and the emitted wave trains, whose maxima are indicated by the lines that emerge from the defect core, will therefore exhibit phase fronts that travel with the group velocities of the asymptotic wave trains away from the core towards $\pm \infty$. The right panel illustrates the profile of the anticipated phase function $\phi(x,t)$.
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