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doi: 10.3934/dcds.2021008

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

Yohei Yamazaki

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

Received  April 2020 Revised  November 2020 Published  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].

Keywords: center stable manifolds, Zakharov–Kuznetsov equation, line solitary wave, transverse instability.
Mathematics Subject Classification: Primary: 35B35, 35Q53.
Citation: Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

R. Côte, C. Muñoz, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

H. Iwasaki, S. 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.  Google Scholar

[11]

J. Jin, Z. 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.  Google Scholar

[12]

J. Jin, Z. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Krieger, K. 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.  Google Scholar

[16]

J. Krieger, K. 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.  Google Scholar

[17]

E. Kirr, P. 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.  Google Scholar

[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.  Google Scholar

[19]

F. Linares, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Y. Martel, F. Merle, K. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

L. Molinet, J.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

R. Côte, C. Muñoz, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

H. Iwasaki, S. 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.  Google Scholar

[11]

J. Jin, Z. 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.  Google Scholar

[12]

J. Jin, Z. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Krieger, K. 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.  Google Scholar

[16]

J. Krieger, K. 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.  Google Scholar

[17]

E. Kirr, P. 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.  Google Scholar

[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.  Google Scholar

[19]

F. Linares, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Y. Martel, F. Merle, K. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

L. Molinet, J.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

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