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March  2017, 16(2): 393-416. doi: 10.3934/cpaa.2017020

Long-term stability for kdv solitons in weighted Hs spaces

Brian Pigott 1, and  Sarah Raynor 2,,

1. 

Department of Mathematics Wofford College, 429 North Church Street, Spartanburg, SC 29303
2. 

Department of Mathematics and Statistics, Wake Forest University, P.O. Box 7388, Winston Salem, NC 27109

* Corresponding author: Sarah Raynor.

Received  December 2015 Revised  October 2016 Published  January 2017

In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the I-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.

Keywords: Korteweg-de Vries equation, soliton, asymptotic stability, traveling waves, stability.
Mathematics Subject Classification: Primary: 35Q53, 35B35; Secondary: 37K40, 35B40, 37K45, 35Q51.
Citation: Brian Pigott, Sarah Raynor. Long-term stability for kdv solitons in weighted Hs spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 393-416. doi: 10.3934/cpaa.2017020
References:
[1]

T. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074.  Google Scholar

[2]

J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. (London) Ser. A, 344 (1975), 363-374.  doi: 10.1098/rspa.1975.0106.  Google Scholar

[3]

J. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. (London) Ser. A, 411 (1987), 395-412.  doi: 10.1098/rspa.1987.0073.  Google Scholar

[4]

T. Buckmaster and H. Koch, The Korteweg-de Vries equation at H-1 regularity, arXiv: 1112.4657 Google Scholar

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Fund. Anal., 211 (2004), 173-218.   Google Scholar

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the instability of the nonlinear Schrodinger equation below the energy norm, Commun. Pure. Appl. Anal., 2 (2003), 33-50.   Google Scholar

[7]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54.   Google Scholar

[8]

Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901.   Google Scholar

[9]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464.   Google Scholar

[10]

Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., 157 (2001), 219-254.  doi: 10.1007/s002050100138.  Google Scholar

[11]

Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations, revisited, Nonlinearity, 18 (2005), 391-427.  doi: 10.1088/0951-7715/18/1/004.  Google Scholar

[12]

Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341 (2008), 391-427.  doi: 10.1007/s00208-007-0194-z.  Google Scholar

[13]

F. Merle and L. Vega, L2 stability of solitons for the KdV equation, Int. Math. Res. Not., 13 (2003), 735-753.  doi: 10.1155/S1073792803208060.  Google Scholar

[14]

T. Mizumachi, Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., 32 (2001), 1050-1080.   Google Scholar

[15]

T. Mizumachi and N. Tzvetkov, L2-stability of solitary waves for the KdV equation via Pego and Weinstein's method, preprint, arXiv: 1403.5321. Google Scholar

[16]

L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 50 (2001), 1745-1776.   Google Scholar

[17]

L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 37 (2002), 1979-2005.   Google Scholar

[18]

R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 2 (1994), 305-349.   Google Scholar

[19]

B. Pigott, Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations, Commun. Pure. Appl. Anal., 13 (2014), 389-418.  doi: 10.3934/cpaa.2014.13.389.  Google Scholar

[20]

B. Pigott and S. Raynor, Asymptotic stability for KdV solitons in weighted spaces via iteration, Submitted, (2013). Google Scholar

[21]

S. Raynor and G. Staffilani, Low regularity stability of solitons for the KdV equation, Commun. Pure. Appl. Anal., 2 (2003), 277-296.  doi: 10.3934/cpaa.2003.2.277.  Google Scholar

[22]

M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math, 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.  Google Scholar

show all references

References:
[1]

T. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074.  Google Scholar

[2]

J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. (London) Ser. A, 344 (1975), 363-374.  doi: 10.1098/rspa.1975.0106.  Google Scholar

[3]

J. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. (London) Ser. A, 411 (1987), 395-412.  doi: 10.1098/rspa.1987.0073.  Google Scholar

[4]

T. Buckmaster and H. Koch, The Korteweg-de Vries equation at H-1 regularity, arXiv: 1112.4657 Google Scholar

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Fund. Anal., 211 (2004), 173-218.   Google Scholar

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the instability of the nonlinear Schrodinger equation below the energy norm, Commun. Pure. Appl. Anal., 2 (2003), 33-50.   Google Scholar

[7]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54.   Google Scholar

[8]

Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901.   Google Scholar

[9]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464.   Google Scholar

[10]

Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., 157 (2001), 219-254.  doi: 10.1007/s002050100138.  Google Scholar

[11]

Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations, revisited, Nonlinearity, 18 (2005), 391-427.  doi: 10.1088/0951-7715/18/1/004.  Google Scholar

[12]

Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341 (2008), 391-427.  doi: 10.1007/s00208-007-0194-z.  Google Scholar

[13]

F. Merle and L. Vega, L2 stability of solitons for the KdV equation, Int. Math. Res. Not., 13 (2003), 735-753.  doi: 10.1155/S1073792803208060.  Google Scholar

[14]

T. Mizumachi, Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., 32 (2001), 1050-1080.   Google Scholar

[15]

T. Mizumachi and N. Tzvetkov, L2-stability of solitary waves for the KdV equation via Pego and Weinstein's method, preprint, arXiv: 1403.5321. Google Scholar

[16]

L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 50 (2001), 1745-1776.   Google Scholar

[17]

L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 37 (2002), 1979-2005.   Google Scholar

[18]

R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 2 (1994), 305-349.   Google Scholar

[19]

B. Pigott, Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations, Commun. Pure. Appl. Anal., 13 (2014), 389-418.  doi: 10.3934/cpaa.2014.13.389.  Google Scholar

[20]

B. Pigott and S. Raynor, Asymptotic stability for KdV solitons in weighted spaces via iteration, Submitted, (2013). Google Scholar

[21]

S. Raynor and G. Staffilani, Low regularity stability of solitons for the KdV equation, Commun. Pure. Appl. Anal., 2 (2003), 277-296.  doi: 10.3934/cpaa.2003.2.277.  Google Scholar

[22]

M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math, 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.  Google Scholar

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