April  2022, 42(4): 1767-1799. doi: 10.3934/dcds.2021171

Instability of the soliton for the focusing, mass-critical generalized KdV equation

Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

Received  March 2021 Published  April 2022 Early access  November 2021

Fund Project: The first author acknowledges the support of NSF grant DMS-1764358

In this paper we prove instability of the soliton for the focusing, mass-critical generalized KdV equation. We prove that the solution to the generalized KdV equation for any initial data with mass smaller than the mass of the soliton and close to the soliton in $ L^{2} $ norm must eventually move away from the soliton.

Citation: Benjamin Dodson, Cristian Gavrus. Instability of the soliton for the focusing, mass-critical generalized KdV equation. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1767-1799. doi: 10.3934/dcds.2021171
References:
[1]

J. Bourgain and W. Wang, Construction of blowup solutions for the nonlinear schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 197-215. 

[2]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in ${H}^{s}$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.

[3]

M. ChristM. J. T. Colliander and M. J. T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, American Journal of Mathematics, 125 (2003), 1235-1293. 

[4]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.  doi: 10.1016/j.aim.2015.04.030.

[5]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when d = 1, Amer. J. Math., 138 (2016), 531-569.  doi: 10.1353/ajm.2016.0016.

[6]

B. Dodson, Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation, Annals of PDE, 3 (2017), 35 pp. doi: 10.1007/s40818-017-0025-9.

[7] P. Drazin, Solitons, Cambridge University Press, 1983.  doi: 10.1017/CBO9780511662843.
[8]

C. Fan, The $L^2$ Weak Sequential Convergence of Radial Focusing Mass Critical NLS Solutions with Mass Above the Ground State, Int. Math. Res. Not. IMRN, 2021 (2021), 4864-4906.  doi: 10.1093/imrn/rny164.

[9]

C. E. 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.

[10]

R. KillipS. KwonS. Shao and M. Visan, On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst., 32 (2012), 191-221.  doi: 10.3934/dcds.2012.32.191.

[11]

Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg–de Vries equation, J. Math. Pures Appl., 79 (2000), 339-425.  doi: 10.1016/S0021-7824(00)00159-8.

[12]

Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg de Vries equation, Geom. Funct. Anal., 11 (2001), 74-123.  doi: 10.1007/PL00001673.

[13]

Y. Martel and F. Merle, Blow up in finite time and dynamics of blow up solutions for the $L^2$–critical generalized KdV equation, J. Amer. Math. Soc., 15 (2002), 617-664.  doi: 10.1090/S0894-0347-02-00392-2.

[14]

Y. Martel and F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized kdv equation, Ann. of Math., 155 (2002), 235-280.  doi: 10.2307/3062156.

[15]

Y. MartelF. Merle and P. Raphaël, Blow up for the critical generalized Korteweg–de Vries equation. i: Dynamics near the soliton, Acta Math., 212 (2014), 59-140.  doi: 10.1007/s11511-014-0109-2.

[16]

Y. MartelF. Merle and P. Raphaël, Blow up for the critical gKdV equation. II: Minimal mass dynamics, J. Eur. Math. Soc. (JEMS), 17 (2015), 1855-1925.  doi: 10.4171/JEMS/547.

[17]

F. Merle, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, Proceedings of the International Congress of Mathematicians, 3 (1998), 57-66. 

[18]

F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14 (2001), 555-578.  doi: 10.1090/S0894-0347-01-00369-1.

[19]

M. Schechter, Spectra of Partial Differential Operators, 2$^{nd}$ edition, North-Holland Series in Applied Mathematics and Mechanics, 14. North-Holland Publishing Co., Amsterdam, 1986.

[20]

S. Shao, The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality, Anal. PDE, 2 (2009), 83-117.  doi: 10.2140/apde.2009.2.83.

[21]

T. Tao, Two remarks on the generalised Korteweg de-Vries equation, Discrete Contin. Dyn. Syst., 18 (2007), 1-14.  doi: 10.3934/dcds.2007.18.1.

[22] E. C. Titchmarsh, Elgenfunction Expansions Associated With Second Order Differential Equations, Oxford, at the Clarendon Press, 1946. 
[23]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. 

[24]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

show all references

References:
[1]

J. Bourgain and W. Wang, Construction of blowup solutions for the nonlinear schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 197-215. 

[2]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in ${H}^{s}$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.

[3]

M. ChristM. J. T. Colliander and M. J. T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, American Journal of Mathematics, 125 (2003), 1235-1293. 

[4]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.  doi: 10.1016/j.aim.2015.04.030.

[5]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when d = 1, Amer. J. Math., 138 (2016), 531-569.  doi: 10.1353/ajm.2016.0016.

[6]

B. Dodson, Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation, Annals of PDE, 3 (2017), 35 pp. doi: 10.1007/s40818-017-0025-9.

[7] P. Drazin, Solitons, Cambridge University Press, 1983.  doi: 10.1017/CBO9780511662843.
[8]

C. Fan, The $L^2$ Weak Sequential Convergence of Radial Focusing Mass Critical NLS Solutions with Mass Above the Ground State, Int. Math. Res. Not. IMRN, 2021 (2021), 4864-4906.  doi: 10.1093/imrn/rny164.

[9]

C. E. 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.

[10]

R. KillipS. KwonS. Shao and M. Visan, On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst., 32 (2012), 191-221.  doi: 10.3934/dcds.2012.32.191.

[11]

Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg–de Vries equation, J. Math. Pures Appl., 79 (2000), 339-425.  doi: 10.1016/S0021-7824(00)00159-8.

[12]

Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg de Vries equation, Geom. Funct. Anal., 11 (2001), 74-123.  doi: 10.1007/PL00001673.

[13]

Y. Martel and F. Merle, Blow up in finite time and dynamics of blow up solutions for the $L^2$–critical generalized KdV equation, J. Amer. Math. Soc., 15 (2002), 617-664.  doi: 10.1090/S0894-0347-02-00392-2.

[14]

Y. Martel and F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized kdv equation, Ann. of Math., 155 (2002), 235-280.  doi: 10.2307/3062156.

[15]

Y. MartelF. Merle and P. Raphaël, Blow up for the critical generalized Korteweg–de Vries equation. i: Dynamics near the soliton, Acta Math., 212 (2014), 59-140.  doi: 10.1007/s11511-014-0109-2.

[16]

Y. MartelF. Merle and P. Raphaël, Blow up for the critical gKdV equation. II: Minimal mass dynamics, J. Eur. Math. Soc. (JEMS), 17 (2015), 1855-1925.  doi: 10.4171/JEMS/547.

[17]

F. Merle, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, Proceedings of the International Congress of Mathematicians, 3 (1998), 57-66. 

[18]

F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14 (2001), 555-578.  doi: 10.1090/S0894-0347-01-00369-1.

[19]

M. Schechter, Spectra of Partial Differential Operators, 2$^{nd}$ edition, North-Holland Series in Applied Mathematics and Mechanics, 14. North-Holland Publishing Co., Amsterdam, 1986.

[20]

S. Shao, The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality, Anal. PDE, 2 (2009), 83-117.  doi: 10.2140/apde.2009.2.83.

[21]

T. Tao, Two remarks on the generalised Korteweg de-Vries equation, Discrete Contin. Dyn. Syst., 18 (2007), 1-14.  doi: 10.3934/dcds.2007.18.1.

[22] E. C. Titchmarsh, Elgenfunction Expansions Associated With Second Order Differential Equations, Oxford, at the Clarendon Press, 1946. 
[23]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. 

[24]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

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