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August  2021, 41(8): 3579-3614. doi: 10.3934/dcds.2021008

Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed

Hirhoshima University, 1-3-2 Kagamiyama, Higashi-Hiroshima City, Hiroshima, 739-8511, Japan

Received  April 2020 Revised  November 2020 Published  August 2021 Early access  January 2021

Fund Project: The first author is supported by JSPS Research Fellowships for Young Scientists under Grant 18J00947

In this paper, we construct center stable manifolds around unstable line solitary waves of the Zakharov–Kuznetsov equation on two dimensional cylindrical spaces $ \mathbb {R} \times \mathbb {T}_L $ ($ {\mathbb T}_L = {\mathbb R}/2\pi L {\mathbb Z} $). In the paper [39], center stable manifolds around unstable line solitary waves have been constructed excluding the case of critical speeds $ c \in \{4n^2/5L^2;n \in {\mathbb Z}, n>1\} $. In the case of critical speeds $ c $, any neighborhood of the line solitary wave with speed $ c $ in the energy space includes solitary waves which are depend on the direction $ {\mathbb T}_L $. To construct center stable manifolds including their solitary waves and to recover the degeneracy of the linearized operator around line solitary waves with critical speed, we prove the stability condition of the center stable manifold for critical speed by applying to the estimate of the 4th order term of a Lyapunov function in [37] and [38].

Citation: Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008
References:
[1]

P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, in Dynam. Report Ser. Dynam. Systems Appl, (eds. U. Kirchgraber and H. O. Walther), Wiley, Chichester, UK, 2 (1989), 1–38.

[2]

M. Beceanu, A critical center-stable manifold for Schrödinger's equation in three dimensions, Comm. Pure Appl. Math., 65 (2012), 431-507.  doi: 10.1002/cpa.21387.

[3]

M. Beceanu, A center-stable manifold for the energy-critical wave equation in $ {\mathbb R}^3$ in the symmetric setting, J. Hyperbolic Differ. Equ., 11 (2014), 437-476.  doi: 10.1142/S021989161450012X.

[4]

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

[5]

T. J. Bridges, Universal geometric conditions for the transverse instability of solitary waves, Phys. Rev. Lett., 84 (2000), 2614-2617. 

[6]

A. Comech and D. E. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.

[7]

R. CôteC. MuñozD. Pilod and G. Simpson, Asymptotic stability of high-dimensional Zakharov–Kuznetsov solitons, Arch. Ration. Mech. Anal., 220 (2016), 639-710.  doi: 10.1007/s00205-015-0939-x.

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[9]

A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 89-112.  doi: 10.1017/S0308210500030614.

[10]

H. IwasakiS. Toh and T. Kawahara, Cylindrical quasi-solitons of the Zakharov-Kuznetsov equation, Physica D Nonlinear Phenomena, 43 (1990), 293-303.  doi: 10.1016/0167-2789(90)90138-F.

[11]

J. JinZ. Lin and C. Zeng, Invariant manifolds of traveling waves of the 3D Gross–Pitaevskii equation in the energy space, Comm. Math. Phys., 364 (2018), 981-1039.  doi: 10.1007/s00220-018-3189-6.

[12]

J. JinZ. Lin and C. Zeng, Dynamics near the solitary waves of the supercritical gKDV equations, J. Differential Equations, 267 (2019), 7213-7262.  doi: 10.1016/j.jde.2019.07.019.

[13]

M. A. Johnson, The transverse instability of periodic waves in Zakharov–Kuznetsov type equations, Stud. Appl. Math., 124 (2010), 323-345.  doi: 10.1111/j.1467-9590.2009.00473.x.

[14]

J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Amer. Math. Soc., 19 (2006), 815-920.  doi: 10.1090/S0894-0347-06-00524-8.

[15]

J. KriegerK. Nakanishi and W. Schlag, Threshold phenomenon for the quintic wave equation in three dimensions, Comm. Math. Phys., 327 (2014), 309-332.  doi: 10.1007/s00220-014-1900-9.

[16]

J. KriegerK. Nakanishi and W. Schlag, Center-stable manifold of the ground state in the energy space for the critical wave equation, Math. Ann., 361 (2015), 1-50.  doi: 10.1007/s00208-014-1059-x.

[17]

E. KirrP. G. Kevrekidis and D. E. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.  doi: 10.1007/s00220-011-1361-3.

[18]

D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov–Kuznetsov equation, in Studies in Phase Space Analysis with Applications to PDEs, Progr. Nonlinear Differential Equations Appl. (eds. M. Cicognani, F. Colombini, and D. Del Santo), New York: Birkhauser/Springer, 84 (2013), 181–213. doi: 10.1007/978-1-4614-6348-1_10.

[19]

F. LinaresA. Pastor and J.-C. Saut, Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton, Comm. Partial Differential Equations, 35 (2010), 1674-1689.  doi: 10.1080/03605302.2010.494195.

[20]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov–Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565.  doi: 10.3934/dcds.2009.24.547.

[21]

M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., 263 (2012), 511-528.  doi: 10.1016/j.jfa.2012.04.006.

[22]

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.

[23]

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.

[24]

Y. MartelF. MerleK. Nakanishi and P. Raphaël, Codimension one threshold manifold for the critical gKdV equation, Comm. Math. Phys., 342 (2016), 1075-1106.  doi: 10.1007/s00220-015-2509-3.

[25]

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.  doi: 10.1137/S0036141098346827.

[26]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Lineaire, 32 (2015), 347-371.  doi: 10.1016/j.anihpc.2013.12.003.

[27]

L. MolinetJ.-C. Saut and N. Tzvetkov, Global well-posedness for the KP-II equation on the background of a non-localized solution, Ann. Inst. H. Poincaré Anal. Non Lineaire, 28 (2011), 653-676.  doi: 10.1016/j.anihpc.2011.04.004.

[28]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations, 44 (2012), 1-45.  doi: 10.1007/s00526-011-0424-9.

[29]

K. Nakanishi and W. Schlag, Invariant manifolds around soliton manifolds for the nonlinear Klein–Gordon equation, SIAM J. Math. Anal., 44 (2012), 1175-1210.  doi: 10.1137/11082720X.

[30]

R. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. 

[31]

D. Pelinovsky, Normal form for transverse instability of the line soliton with a nearly critical speed of propagation, Math. Model. Nat. Phenom., 13 (2018), 1-20.  doi: 10.1051/mmnp/2018024.

[32]

F. Ribaud and S. Vento, Well-posedness results for the three-dimensional Zakharov–Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.  doi: 10.1137/110850566.

[33]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures. Appl., 90 (2008), 550-590.  doi: 10.1016/j.matpur.2008.07.004.

[34]

W. Schlag, Stable manifolds for an orbitally unstable nonlinear Schrödinger equation, Ann. of Math. (2), 169 (2009), 139-227.  doi: 10.4007/annals.2009.169.139.

[35]

B. K. Shivamoggi, The painlevé analysis of the Zakharov–Kuznetsov equation, Phys. Scripta, 42 (1990), 641-642.  doi: 10.1088/0031-8949/42/6/001.

[36]

J. Villarroel and M. J. Ablowitz, On the initial value problem for the KPII equation with data that do not decay along a line, Nonlinearity, 17 (2004), 1843-1866.  doi: 10.1088/0951-7715/17/5/015.

[37]

Y. Yamazaki, Stability of line standing waves near the bifurcation point for nonlinear Schrödinger equations, Kodai Math. J., 38 (2015), 65-96.  doi: 10.2996/kmj/1426684443.

[38]

Y. Yamazaki, Stability for line solitary waves of Zakharov–Kuznetsov equation, J. Differential Equations, 262 (2017), 4336-4389.  doi: 10.1016/j.jde.2017.01.006.

[39]

Y. Yamazaki, Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation, arXiv: 1808.07315.

[40]

V. E. Zakharov and E. A. Kuznetsov, On three dimensional solitons, Sov. Phys.-JETP, 39 (1974), 285-286. 

show all references

References:
[1]

P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, in Dynam. Report Ser. Dynam. Systems Appl, (eds. U. Kirchgraber and H. O. Walther), Wiley, Chichester, UK, 2 (1989), 1–38.

[2]

M. Beceanu, A critical center-stable manifold for Schrödinger's equation in three dimensions, Comm. Pure Appl. Math., 65 (2012), 431-507.  doi: 10.1002/cpa.21387.

[3]

M. Beceanu, A center-stable manifold for the energy-critical wave equation in $ {\mathbb R}^3$ in the symmetric setting, J. Hyperbolic Differ. Equ., 11 (2014), 437-476.  doi: 10.1142/S021989161450012X.

[4]

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

[5]

T. J. Bridges, Universal geometric conditions for the transverse instability of solitary waves, Phys. Rev. Lett., 84 (2000), 2614-2617. 

[6]

A. Comech and D. E. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.

[7]

R. CôteC. MuñozD. Pilod and G. Simpson, Asymptotic stability of high-dimensional Zakharov–Kuznetsov solitons, Arch. Ration. Mech. Anal., 220 (2016), 639-710.  doi: 10.1007/s00205-015-0939-x.

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[9]

A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 89-112.  doi: 10.1017/S0308210500030614.

[10]

H. IwasakiS. Toh and T. Kawahara, Cylindrical quasi-solitons of the Zakharov-Kuznetsov equation, Physica D Nonlinear Phenomena, 43 (1990), 293-303.  doi: 10.1016/0167-2789(90)90138-F.

[11]

J. JinZ. Lin and C. Zeng, Invariant manifolds of traveling waves of the 3D Gross–Pitaevskii equation in the energy space, Comm. Math. Phys., 364 (2018), 981-1039.  doi: 10.1007/s00220-018-3189-6.

[12]

J. JinZ. Lin and C. Zeng, Dynamics near the solitary waves of the supercritical gKDV equations, J. Differential Equations, 267 (2019), 7213-7262.  doi: 10.1016/j.jde.2019.07.019.

[13]

M. A. Johnson, The transverse instability of periodic waves in Zakharov–Kuznetsov type equations, Stud. Appl. Math., 124 (2010), 323-345.  doi: 10.1111/j.1467-9590.2009.00473.x.

[14]

J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Amer. Math. Soc., 19 (2006), 815-920.  doi: 10.1090/S0894-0347-06-00524-8.

[15]

J. KriegerK. Nakanishi and W. Schlag, Threshold phenomenon for the quintic wave equation in three dimensions, Comm. Math. Phys., 327 (2014), 309-332.  doi: 10.1007/s00220-014-1900-9.

[16]

J. KriegerK. Nakanishi and W. Schlag, Center-stable manifold of the ground state in the energy space for the critical wave equation, Math. Ann., 361 (2015), 1-50.  doi: 10.1007/s00208-014-1059-x.

[17]

E. KirrP. G. Kevrekidis and D. E. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.  doi: 10.1007/s00220-011-1361-3.

[18]

D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov–Kuznetsov equation, in Studies in Phase Space Analysis with Applications to PDEs, Progr. Nonlinear Differential Equations Appl. (eds. M. Cicognani, F. Colombini, and D. Del Santo), New York: Birkhauser/Springer, 84 (2013), 181–213. doi: 10.1007/978-1-4614-6348-1_10.

[19]

F. LinaresA. Pastor and J.-C. Saut, Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton, Comm. Partial Differential Equations, 35 (2010), 1674-1689.  doi: 10.1080/03605302.2010.494195.

[20]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov–Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565.  doi: 10.3934/dcds.2009.24.547.

[21]

M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., 263 (2012), 511-528.  doi: 10.1016/j.jfa.2012.04.006.

[22]

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.

[23]

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.

[24]

Y. MartelF. MerleK. Nakanishi and P. Raphaël, Codimension one threshold manifold for the critical gKdV equation, Comm. Math. Phys., 342 (2016), 1075-1106.  doi: 10.1007/s00220-015-2509-3.

[25]

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.  doi: 10.1137/S0036141098346827.

[26]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Lineaire, 32 (2015), 347-371.  doi: 10.1016/j.anihpc.2013.12.003.

[27]

L. MolinetJ.-C. Saut and N. Tzvetkov, Global well-posedness for the KP-II equation on the background of a non-localized solution, Ann. Inst. H. Poincaré Anal. Non Lineaire, 28 (2011), 653-676.  doi: 10.1016/j.anihpc.2011.04.004.

[28]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations, 44 (2012), 1-45.  doi: 10.1007/s00526-011-0424-9.

[29]

K. Nakanishi and W. Schlag, Invariant manifolds around soliton manifolds for the nonlinear Klein–Gordon equation, SIAM J. Math. Anal., 44 (2012), 1175-1210.  doi: 10.1137/11082720X.

[30]

R. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. 

[31]

D. Pelinovsky, Normal form for transverse instability of the line soliton with a nearly critical speed of propagation, Math. Model. Nat. Phenom., 13 (2018), 1-20.  doi: 10.1051/mmnp/2018024.

[32]

F. Ribaud and S. Vento, Well-posedness results for the three-dimensional Zakharov–Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.  doi: 10.1137/110850566.

[33]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures. Appl., 90 (2008), 550-590.  doi: 10.1016/j.matpur.2008.07.004.

[34]

W. Schlag, Stable manifolds for an orbitally unstable nonlinear Schrödinger equation, Ann. of Math. (2), 169 (2009), 139-227.  doi: 10.4007/annals.2009.169.139.

[35]

B. K. Shivamoggi, The painlevé analysis of the Zakharov–Kuznetsov equation, Phys. Scripta, 42 (1990), 641-642.  doi: 10.1088/0031-8949/42/6/001.

[36]

J. Villarroel and M. J. Ablowitz, On the initial value problem for the KPII equation with data that do not decay along a line, Nonlinearity, 17 (2004), 1843-1866.  doi: 10.1088/0951-7715/17/5/015.

[37]

Y. Yamazaki, Stability of line standing waves near the bifurcation point for nonlinear Schrödinger equations, Kodai Math. J., 38 (2015), 65-96.  doi: 10.2996/kmj/1426684443.

[38]

Y. Yamazaki, Stability for line solitary waves of Zakharov–Kuznetsov equation, J. Differential Equations, 262 (2017), 4336-4389.  doi: 10.1016/j.jde.2017.01.006.

[39]

Y. Yamazaki, Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation, arXiv: 1808.07315.

[40]

V. E. Zakharov and E. A. Kuznetsov, On three dimensional solitons, Sov. Phys.-JETP, 39 (1974), 285-286. 

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