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Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift
Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities
Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, 7491 Trondheim, Norway |
$ \begin{equation*} u_t+ (Lu - n(u))_x = 0\,, \end{equation*} $ |
$ L $ |
$ n $ |
$ m $ |
$ L $ |
$ s > 0 $ |
$ n $ |
$ H^{1+s} $ |
References:
[1] |
J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, in Applied Analysis, Contemp. Math., 221, Amer. Math. Soc., Providence, RI, 1999, 1–29.
doi: 10.1090/conm/221/03116. |
[2] |
J. P. Albert, J. L. Bona and J.-C. Saut,
Model equations for waves in stratified fluids, Proc. Roy. Soc. London Ser. A, 453 (1997), 1233-1260.
doi: 10.1098/rspa.1997.0068. |
[3] |
M. N. Arnesen,
Existence of solitary-wave solutions to nonlocal equations, Discrete Contin. Dyn. Syst., 36 (2016), 3483-3510.
doi: 10.3934/dcds.2016.36.3483. |
[4] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/978-3-642-66451-9. |
[5] |
G. Bourdaud, M. Moussai and W. Sickel,
Composition operators acting on Besov spaces on the real line, Ann. Mat. Pura Appl. (4), 193 (2014), 1519-1554.
doi: 10.1007/s10231-013-0342-x. |
[6] |
H. Chen and J. L. Bona,
Periodic traveling-wave solutions of nonlinear dispersive evolution equations, Discrete Contin. Dyn. Syst., 33 (2013), 4841-4873.
doi: 10.3934/dcds.2013.33.4841. |
[7] |
M. Ehrnström, M. D. Groves and E. Wahlén,
On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 2903-2936.
doi: 10.1088/0951-7715/25/10/2903. |
[8] |
M. Ehrnström, M. A. Johnson, O. I. H. Maehlen and F. Remonato,
On the bifurcation diagram of the capillary–gravity Whitham equation, Water Waves, 1 (2019), 275-313.
doi: 10.1007/s42286-019-00019-4. |
[9] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\Bbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[10] |
M. D. Groves and E. Wahlén,
On the existence and conditional energetic stability of solitary gravity-capillary surface waves on deep water, J. Math. Fluid Mech., 13 (2011), 593-627.
doi: 10.1007/s00021-010-0034-x. |
[11] |
M. A. Johnson and J. D. Wright,
Generalized solitary waves in the gravity-capillary Whitham equation, Studies in Applied Mathematics, 144 (2020), 102-130.
doi: 10.1111/sapm.12288. |
[12] |
D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, 188, American Mathematical Society, Providence, RI, 2013.
doi: 10.1090/surv/188. |
[13] |
F. Linares, D. Pilod and J.-C. Saut,
Dispersive perturbations of Burgers and hyperbolic equations. I: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.
doi: 10.1137/130912001. |
[14] |
F. Linares, D. Pilod and J.-C. Saut,
Remarks on the orbital stability of ground state solutions of fKdV and related equations, Adv. Differential Equations, 20 (2015), 835-858.
|
[15] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[16] |
M. Moussai,
Composition operators on Besov spaces in the limiting case $s=1+1/p$, Studia Math., 241 (2018), 1-15.
doi: 10.4064/sm8136-4-2017. |
[17] |
S. M. Sun,
Non-existence of truly solitary waves in water with small surface tension, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2191-2228.
doi: 10.1098/rspa.1999.0399. |
[18] |
M. I. Weinstein,
Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, 12 (1987), 1133-1173.
doi: 10.1080/03605308708820522. |
[19] |
L. Zeng,
Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type, J. Differential Equations, 188 (2003), 1-32.
doi: 10.1016/S0022-0396(02)00061-X. |
show all references
References:
[1] |
J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, in Applied Analysis, Contemp. Math., 221, Amer. Math. Soc., Providence, RI, 1999, 1–29.
doi: 10.1090/conm/221/03116. |
[2] |
J. P. Albert, J. L. Bona and J.-C. Saut,
Model equations for waves in stratified fluids, Proc. Roy. Soc. London Ser. A, 453 (1997), 1233-1260.
doi: 10.1098/rspa.1997.0068. |
[3] |
M. N. Arnesen,
Existence of solitary-wave solutions to nonlocal equations, Discrete Contin. Dyn. Syst., 36 (2016), 3483-3510.
doi: 10.3934/dcds.2016.36.3483. |
[4] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/978-3-642-66451-9. |
[5] |
G. Bourdaud, M. Moussai and W. Sickel,
Composition operators acting on Besov spaces on the real line, Ann. Mat. Pura Appl. (4), 193 (2014), 1519-1554.
doi: 10.1007/s10231-013-0342-x. |
[6] |
H. Chen and J. L. Bona,
Periodic traveling-wave solutions of nonlinear dispersive evolution equations, Discrete Contin. Dyn. Syst., 33 (2013), 4841-4873.
doi: 10.3934/dcds.2013.33.4841. |
[7] |
M. Ehrnström, M. D. Groves and E. Wahlén,
On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 2903-2936.
doi: 10.1088/0951-7715/25/10/2903. |
[8] |
M. Ehrnström, M. A. Johnson, O. I. H. Maehlen and F. Remonato,
On the bifurcation diagram of the capillary–gravity Whitham equation, Water Waves, 1 (2019), 275-313.
doi: 10.1007/s42286-019-00019-4. |
[9] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\Bbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[10] |
M. D. Groves and E. Wahlén,
On the existence and conditional energetic stability of solitary gravity-capillary surface waves on deep water, J. Math. Fluid Mech., 13 (2011), 593-627.
doi: 10.1007/s00021-010-0034-x. |
[11] |
M. A. Johnson and J. D. Wright,
Generalized solitary waves in the gravity-capillary Whitham equation, Studies in Applied Mathematics, 144 (2020), 102-130.
doi: 10.1111/sapm.12288. |
[12] |
D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, 188, American Mathematical Society, Providence, RI, 2013.
doi: 10.1090/surv/188. |
[13] |
F. Linares, D. Pilod and J.-C. Saut,
Dispersive perturbations of Burgers and hyperbolic equations. I: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.
doi: 10.1137/130912001. |
[14] |
F. Linares, D. Pilod and J.-C. Saut,
Remarks on the orbital stability of ground state solutions of fKdV and related equations, Adv. Differential Equations, 20 (2015), 835-858.
|
[15] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[16] |
M. Moussai,
Composition operators on Besov spaces in the limiting case $s=1+1/p$, Studia Math., 241 (2018), 1-15.
doi: 10.4064/sm8136-4-2017. |
[17] |
S. M. Sun,
Non-existence of truly solitary waves in water with small surface tension, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2191-2228.
doi: 10.1098/rspa.1999.0399. |
[18] |
M. I. Weinstein,
Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, 12 (1987), 1133-1173.
doi: 10.1080/03605308708820522. |
[19] |
L. Zeng,
Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type, J. Differential Equations, 188 (2003), 1-32.
doi: 10.1016/S0022-0396(02)00061-X. |
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