doi: 10.3934/dcdss.2022070
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Large-amplitude modulation of periodic traveling waves

1. 

Université Bordeaux 1, Talence, France

2. 

Indiana University, Bloomington, IN 47405, USA

* Corresponding author: Kevin Zumbrun

Received  November 2021 Revised  February 2022 Early access March 2022

Fund Project: Research of K.Z. was partially supported under NSF grant no. DMS-0300487

We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.

Citation: Guy Métivier, Kevin Zumbrun. Large-amplitude modulation of periodic traveling waves. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022070
References:
[1]

M. BeckT. T. NguyenB. 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.

[2]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sc. E.N.S. Paris, 14 (1981), 209-246.  doi: 10.24033/asens.1404.

[3]

P. Collet and J.-P. Eckmann, The time-dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys., 132 (1990), 139-153.  doi: 10.1007/BF02278004.

[4]

J.-F. Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., 47 (2009), 2844-2871.  doi: 10.1137/080728342.

[5]

A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wavetrains, Mem. Amer. Math. Soc., 199 (2009), viii+105 pp. doi: 10.1090/memo/0934.

[6]

R. A. Gardner, On the structure of the spectra of periodic traveling waves, J. Math. Pures Appl., 72 (1993), 415-439. 

[7]

O. GuèsG. MétivierM. Williams and K Zumbrun, Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Ration. Mech. Anal., 175 (2005), 151-244.  doi: 10.1007/s00205-004-0342-5.

[8]

O. M. I. O. GuèsG. MétivierM. Williams and K. Zumbrun, Navier–Stokes regularization of multidimensional Euler shocks, Ann. Sci. École Norm. Sup., 39 (2006), 75-175.  doi: 10.1016/j.ansens.2005.12.002.

[9]

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.  doi: 10.1002/sapm197756295.

[10]

M. A. JohnsonP. NobleL. 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.

[11]

M. JohnsonP. NobleL. M. Rodrigues and K. Zumbrun, Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation, Arch. Ration. Mech. Anal., 207 (2013), 669-692.  doi: 10.1007/s00205-012-0572-x.

[12]

M. A. JohnsonP. NobleL. M. Rodrigues and K. Zumbrun, Behaviour of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations, Inventiones Math., 197 (2014), 115-213.  doi: 10.1007/s00222-013-0481-0.

[13]

M. A. Johnson and K. Zumbrun, Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 471-483.  doi: 10.1016/j.anihpc.2011.05.003.

[14]

J.-L. JolyG. Métivier and J. Rauch, Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J., 70 (1993), 373-404.  doi: 10.1215/S0012-7094-93-07007-X.

[15]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298.  doi: 10.1002/cpa.3160230304.

[16]

A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 43 (1983), v+93 pp. doi: 10.1090/memo/0281.

[17]

G. Métivier, Stability of multidimensional shocks, Advances in the Theory of Shock Waves, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 47 (2001), 25-103. 

[18]

G. Métivier and K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc., 175 (2005), vi+107 pp. doi: 10.1090/memo/0826.

[19]

P. Noble and L. M. Rodrigues, Whitham's modulation equations and stability of periodic wave solutions of the generalized Kuramoto-Sivashinsky equations, Indiana Univ. Math. J., 62 (2013), 753-783.  doi: 10.1512/iumj.2013.62.4955.

[20]

M. Oh and K. Zumbrun, Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions, Z. Anal. Anwend., 25 (2006), 1-21.  doi: 10.4171/ZAA/1275.

[21]

M. Oh and K. Zumbrun, Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions, Arch. Ration. Mech. Anal., 196 (2010), 1-20.  doi: 10.1007/s00205-009-0229-6.

[22]

B. Sandstede and A. Scheel, Defects in oscillatory media: Toward a classification, SIAM Journal on Applied Dynamical Systems, 3 (2004), 1-68.  doi: 10.1137/030600192.

[23]

B. SandstedeA. ScheelG. Schneider and H. Uecker, Diffusive mixing of periodic wave trains in reaction-diffusion systems, J. Diff. Eq., 252 (2012), 3541-3574.  doi: 10.1016/j.jde.2011.10.014.

[24]

G. Schneider, Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation, Comm. Math. Phys., 178 (1996), 679-702.  doi: 10.1007/BF02108820.

[25]

G. Schneider, Global existence via Ginzburg-Landau formalism and pseudo-orbits of the Ginzburg-Landau approximations, Comm. Math. Phys., 164 (1994), 159-179.  doi: 10.1007/BF02108810.

[26]

D. Serre, Spectral stability of periodic solutions of viscous conservation laws: Large wavelength analysis, Comm. Partial Differential Equations, 30 (2005), 259-282.  doi: 10.1081/PDE-200044492.

[27]

A. van Harten, On the validity of the Ginzburg-Landau's equation, J. Nonlinear Sci., 1 (1991), 397-422.  doi: 10.1007/BF02429847.

[28]

G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1974.

[29]

K. Zumbrun, 2-modified characteristic Fredholm determinants, Hill's method, and the periodic Evans function of Gardner, Z. Anal. Anwend., 31 (2012), 463-472.  doi: 10.4171/ZAA/1469.

show all references

References:
[1]

M. BeckT. T. NguyenB. 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.

[2]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sc. E.N.S. Paris, 14 (1981), 209-246.  doi: 10.24033/asens.1404.

[3]

P. Collet and J.-P. Eckmann, The time-dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys., 132 (1990), 139-153.  doi: 10.1007/BF02278004.

[4]

J.-F. Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., 47 (2009), 2844-2871.  doi: 10.1137/080728342.

[5]

A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wavetrains, Mem. Amer. Math. Soc., 199 (2009), viii+105 pp. doi: 10.1090/memo/0934.

[6]

R. A. Gardner, On the structure of the spectra of periodic traveling waves, J. Math. Pures Appl., 72 (1993), 415-439. 

[7]

O. GuèsG. MétivierM. Williams and K Zumbrun, Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Ration. Mech. Anal., 175 (2005), 151-244.  doi: 10.1007/s00205-004-0342-5.

[8]

O. M. I. O. GuèsG. MétivierM. Williams and K. Zumbrun, Navier–Stokes regularization of multidimensional Euler shocks, Ann. Sci. École Norm. Sup., 39 (2006), 75-175.  doi: 10.1016/j.ansens.2005.12.002.

[9]

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.  doi: 10.1002/sapm197756295.

[10]

M. A. JohnsonP. NobleL. 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.

[11]

M. JohnsonP. NobleL. M. Rodrigues and K. Zumbrun, Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation, Arch. Ration. Mech. Anal., 207 (2013), 669-692.  doi: 10.1007/s00205-012-0572-x.

[12]

M. A. JohnsonP. NobleL. M. Rodrigues and K. Zumbrun, Behaviour of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations, Inventiones Math., 197 (2014), 115-213.  doi: 10.1007/s00222-013-0481-0.

[13]

M. A. Johnson and K. Zumbrun, Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 471-483.  doi: 10.1016/j.anihpc.2011.05.003.

[14]

J.-L. JolyG. Métivier and J. Rauch, Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J., 70 (1993), 373-404.  doi: 10.1215/S0012-7094-93-07007-X.

[15]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298.  doi: 10.1002/cpa.3160230304.

[16]

A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 43 (1983), v+93 pp. doi: 10.1090/memo/0281.

[17]

G. Métivier, Stability of multidimensional shocks, Advances in the Theory of Shock Waves, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 47 (2001), 25-103. 

[18]

G. Métivier and K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc., 175 (2005), vi+107 pp. doi: 10.1090/memo/0826.

[19]

P. Noble and L. M. Rodrigues, Whitham's modulation equations and stability of periodic wave solutions of the generalized Kuramoto-Sivashinsky equations, Indiana Univ. Math. J., 62 (2013), 753-783.  doi: 10.1512/iumj.2013.62.4955.

[20]

M. Oh and K. Zumbrun, Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions, Z. Anal. Anwend., 25 (2006), 1-21.  doi: 10.4171/ZAA/1275.

[21]

M. Oh and K. Zumbrun, Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions, Arch. Ration. Mech. Anal., 196 (2010), 1-20.  doi: 10.1007/s00205-009-0229-6.

[22]

B. Sandstede and A. Scheel, Defects in oscillatory media: Toward a classification, SIAM Journal on Applied Dynamical Systems, 3 (2004), 1-68.  doi: 10.1137/030600192.

[23]

B. SandstedeA. ScheelG. Schneider and H. Uecker, Diffusive mixing of periodic wave trains in reaction-diffusion systems, J. Diff. Eq., 252 (2012), 3541-3574.  doi: 10.1016/j.jde.2011.10.014.

[24]

G. Schneider, Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation, Comm. Math. Phys., 178 (1996), 679-702.  doi: 10.1007/BF02108820.

[25]

G. Schneider, Global existence via Ginzburg-Landau formalism and pseudo-orbits of the Ginzburg-Landau approximations, Comm. Math. Phys., 164 (1994), 159-179.  doi: 10.1007/BF02108810.

[26]

D. Serre, Spectral stability of periodic solutions of viscous conservation laws: Large wavelength analysis, Comm. Partial Differential Equations, 30 (2005), 259-282.  doi: 10.1081/PDE-200044492.

[27]

A. van Harten, On the validity of the Ginzburg-Landau's equation, J. Nonlinear Sci., 1 (1991), 397-422.  doi: 10.1007/BF02429847.

[28]

G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1974.

[29]

K. Zumbrun, 2-modified characteristic Fredholm determinants, Hill's method, and the periodic Evans function of Gardner, Z. Anal. Anwend., 31 (2012), 463-472.  doi: 10.4171/ZAA/1469.

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