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April  2017, 37(4): 2065-2075. doi: 10.3934/dcds.2017088

Long-time stability of small FPU solitary waves

1. 

Department of Applied Mathematics, Western University, London, ON, N6A 3K7, Canada

2. 

Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Received  February 2016 Revised  November 2016 Published  December 2016

Fund Project: The work of A. Khan was performed during MSc program at McMaster University in 2013-2015. The work of D.E. Pelinovsky is supported by the NSERC grant. The authors thank E. Dumas, T. Penati, and G. Schneider for discussions and collaborations.

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentialsare described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitudeapproximation is usually performed on the time scale, for which dynamics of the KdV equation is defined.We show how to extend justification analysis on longer time intervals provided dynamics of the generalized KdVequation is globally well-posed in Sobolev spaces and either the Sobolev norms are globally boundedor they grow at most polynomially. The time intervals are extended respectively by the logarithmic or double logarithmic factorsin terms of the small amplitude parameter. Controlling the approximation error on longer time intervalsallows us to deduce nonlinear metastability of small FPU solitary waves from orbital stability of the KdV solitary waves.

Citation: Amjad Khan, Dmitry E. Pelinovsky. Long-time stability of small FPU solitary waves. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2065-2075. doi: 10.3934/dcds.2017088
References:
[1]

D. Bambusi and T. Penati, Continuous approximation of breathers in 1D and 2D DNLS lattices, Nonlinearity, 23 (2010), 143-157.  doi: 10.1088/0951-7715/23/1/008.  Google Scholar

[2]

D. Bambusi and A. Ponno, On metastability in FPU, Comm. Math. Phys., 264 (2006), 539-561.  doi: 10.1007/s00220-005-1488-1.  Google Scholar

[3]

G. N. BenesA. Hoffman and C. E. Wayne, Asymptotic stability of the Toda m-soliton, J. Math. Anal. Appl., 386 (2012), 445-460.  doi: 10.1016/j.jmaa.2011.08.007.  Google Scholar

[4]

B. Bidegaray-FesquetE. Dumas and G. James, From Newton's cradle to the discrete p-Schrödinger equation, SIAM J. Math. Anal., 45 (2013), 3404-3430.  doi: 10.1137/130924196.  Google Scholar

[5]

J. BonaY. Liu and N. V. Ngueyn, Stability fo solitary waves in higher-order Sobolev spaces, Comm Math. Sci., 2 (2004), 35-52.  doi: 10.4310/CMS.2004.v2.n1.a3.  Google Scholar

[6]

E. Dumas and D. E. Pelinovsky, Justification of the log-KdV equation in granular chains: The case of precompression, SIAM J. Math. Anal., 46 (2014), 4075-4103.  doi: 10.1137/140969270.  Google Scholar

[7]

G. Friesecke and R. L. Pego, Solitary waves on FPU lattices, Nonlinearity, 12 (1999), 1601-1627; 15 (2002), 1343-1359; 17 (2004), 207-227; 17 (2004), 229-251. Google Scholar

[8]

J. GaisonS. MoskowJ. D. Wright and Q. Zhang, Approximation of polyatomic FPU lattices by KdV equations, Multiscale Model. Simul., 12 (2014), 953-995.  doi: 10.1137/130941638.  Google Scholar

[9]

A. Hoffman and C. E. Wayne, Asymptotic two-soliton solutions in the Fermi-Pasta-Ulam model, J. Dynam. Differential Equations, 21 (2009), 343-351.  doi: 10.1007/s10884-009-9134-9.  Google Scholar

[10]

T. Kato, On the Korteweg-de Vries equation, Manuscript Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.  Google Scholar

[11]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Stud. Appl. Math., 8 (1983), 93-128.   Google Scholar

[12]

C. KenigG. Ponce and L. Vega, Well-posedness of the initial-value problem for the Korteweg-De Vries equation, J. Americ. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[13]

C. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-De Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[14]

D. Lannes and J. Rauch, Validity of nonlinear geometric optics with times growing logarithmically, Proc. AMS, 129 (2000), 1087-1096.  doi: 10.1090/S0002-9939-00-05845-7.  Google Scholar

[15]

R. M. MiuraC. S. Gardner and M. D. Kruskal, Korteweg-de Vries equations and generalization. Ⅱ. Existence of conservation laws and constants of motion, J. Math. Phys., 9 (1968), 1204-1209.  doi: 10.1063/1.1664701.  Google Scholar

[16]

T. Mizumachi, Asymptotic stability of lattice solitons in the energy space, Commun. Math. Phys., 288 (2009), 125-144.  doi: 10.1007/s00220-009-0768-6.  Google Scholar

[17]

T. Mizumachi, Asymptotic stability of N-solitary waves of the FPU lattices, Archive for Rational Mechanics and Analysis, 207 (2013), 393-457.  doi: 10.1007/s00205-012-0564-x.  Google Scholar

[18]

R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London A, 340 (1992), 47-94.  doi: 10.1098/rsta.1992.0055.  Google Scholar

[19]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.  doi: 10.1007/BF02101705.  Google Scholar

[20]

D. PelinovskyT. Penati and S. Paleari, Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrödinger equations, Rev. Math. Phys., 28 (2016), 1650015(25 pages).  doi: 10.1142/S0129055X1650015X.  Google Scholar

[21]

G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, In International Conference on Differential Equations (Berlin, 1999), vol. 1 (eds B Fiedler, K Gröger, J Sprekels), pp. 390-404 (World Sci. Publishing, River Edge, NJ, USA, 2000).  Google Scholar

[22]

G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142.  doi: 10.1215/S0012-7094-97-08604-X.  Google Scholar

show all references

References:
[1]

D. Bambusi and T. Penati, Continuous approximation of breathers in 1D and 2D DNLS lattices, Nonlinearity, 23 (2010), 143-157.  doi: 10.1088/0951-7715/23/1/008.  Google Scholar

[2]

D. Bambusi and A. Ponno, On metastability in FPU, Comm. Math. Phys., 264 (2006), 539-561.  doi: 10.1007/s00220-005-1488-1.  Google Scholar

[3]

G. N. BenesA. Hoffman and C. E. Wayne, Asymptotic stability of the Toda m-soliton, J. Math. Anal. Appl., 386 (2012), 445-460.  doi: 10.1016/j.jmaa.2011.08.007.  Google Scholar

[4]

B. Bidegaray-FesquetE. Dumas and G. James, From Newton's cradle to the discrete p-Schrödinger equation, SIAM J. Math. Anal., 45 (2013), 3404-3430.  doi: 10.1137/130924196.  Google Scholar

[5]

J. BonaY. Liu and N. V. Ngueyn, Stability fo solitary waves in higher-order Sobolev spaces, Comm Math. Sci., 2 (2004), 35-52.  doi: 10.4310/CMS.2004.v2.n1.a3.  Google Scholar

[6]

E. Dumas and D. E. Pelinovsky, Justification of the log-KdV equation in granular chains: The case of precompression, SIAM J. Math. Anal., 46 (2014), 4075-4103.  doi: 10.1137/140969270.  Google Scholar

[7]

G. Friesecke and R. L. Pego, Solitary waves on FPU lattices, Nonlinearity, 12 (1999), 1601-1627; 15 (2002), 1343-1359; 17 (2004), 207-227; 17 (2004), 229-251. Google Scholar

[8]

J. GaisonS. MoskowJ. D. Wright and Q. Zhang, Approximation of polyatomic FPU lattices by KdV equations, Multiscale Model. Simul., 12 (2014), 953-995.  doi: 10.1137/130941638.  Google Scholar

[9]

A. Hoffman and C. E. Wayne, Asymptotic two-soliton solutions in the Fermi-Pasta-Ulam model, J. Dynam. Differential Equations, 21 (2009), 343-351.  doi: 10.1007/s10884-009-9134-9.  Google Scholar

[10]

T. Kato, On the Korteweg-de Vries equation, Manuscript Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.  Google Scholar

[11]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Stud. Appl. Math., 8 (1983), 93-128.   Google Scholar

[12]

C. KenigG. Ponce and L. Vega, Well-posedness of the initial-value problem for the Korteweg-De Vries equation, J. Americ. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[13]

C. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-De Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[14]

D. Lannes and J. Rauch, Validity of nonlinear geometric optics with times growing logarithmically, Proc. AMS, 129 (2000), 1087-1096.  doi: 10.1090/S0002-9939-00-05845-7.  Google Scholar

[15]

R. M. MiuraC. S. Gardner and M. D. Kruskal, Korteweg-de Vries equations and generalization. Ⅱ. Existence of conservation laws and constants of motion, J. Math. Phys., 9 (1968), 1204-1209.  doi: 10.1063/1.1664701.  Google Scholar

[16]

T. Mizumachi, Asymptotic stability of lattice solitons in the energy space, Commun. Math. Phys., 288 (2009), 125-144.  doi: 10.1007/s00220-009-0768-6.  Google Scholar

[17]

T. Mizumachi, Asymptotic stability of N-solitary waves of the FPU lattices, Archive for Rational Mechanics and Analysis, 207 (2013), 393-457.  doi: 10.1007/s00205-012-0564-x.  Google Scholar

[18]

R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London A, 340 (1992), 47-94.  doi: 10.1098/rsta.1992.0055.  Google Scholar

[19]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.  doi: 10.1007/BF02101705.  Google Scholar

[20]

D. PelinovskyT. Penati and S. Paleari, Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrödinger equations, Rev. Math. Phys., 28 (2016), 1650015(25 pages).  doi: 10.1142/S0129055X1650015X.  Google Scholar

[21]

G. Schneider and C. E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, In International Conference on Differential Equations (Berlin, 1999), vol. 1 (eds B Fiedler, K Gröger, J Sprekels), pp. 390-404 (World Sci. Publishing, River Edge, NJ, USA, 2000).  Google Scholar

[22]

G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142.  doi: 10.1215/S0012-7094-97-08604-X.  Google Scholar

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