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Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators
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 |
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 [
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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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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