doi: 10.3934/dcds.2021171
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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 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021171
References:
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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.   Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

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