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Existence of solitary-wave solutions to nonlocal equations
1. | Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway |
References:
[1] |
J. P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, Contemp. Math., 221 (1999), 1-29.
doi: 10.1090/conm/221/03116. |
[2] |
J. P. Albert, J. L. Bona and D. B. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, Phys. D, 24 (1987), 343-366.
doi: 10.1016/0167-2789(87)90084-4. |
[3] |
J. P. Albert, J. L. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. R. Soc. Lond., 453 (1997), 1233-1260.
doi: 10.1098/rspa.1997.0068. |
[4] |
J. Angulo Pava, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, American Mathematical Society, Providence, RI, 2009.
doi: 10.1090/surv/156. |
[5] |
H. Chen and J. L. Bona, Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Differential Equations, 3 (1998), 51-84. |
[6] |
R. R. Coifman and Y. Meyer, Au-delà des Opérateurs Pseudo-différentiels, Astérisque, 1978. |
[7] |
M. Ehrnström, J. Escher and L. Pei, A note on the local well-posedness for the Whitham equation, Elliptic and Parabolic Equations, Springer Proceedings in Mathematics & Statistics, 119 (2015), 63-75.
doi: 10.1007/978-3-319-12547-3_3. |
[8] |
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. |
[9] |
M. Ehrnström and H. Kalisch, Global bifurcation for the Whitham equation, Math. Model. Nat. Phenom., 8 (2013), 13-30.
doi: 10.1051/mmnp/20138502. |
[10] |
R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[11] |
B. Guo and D. Huang, Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys., 53 (2012), 083702, 15pp.
doi: 10.1063/1.4746806. |
[12] |
V. M. Hur and M. A. Johnson, Modulational instability in the Whitham equation for water waves, Studies in Applied Mathematics, 134 (2015), 120-143.
doi: 10.1111/sapm.12061. |
[13] |
C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Physica D, 295/296 (2015), 46-65.
doi: 10.1016/j.physd.2014.12.004. |
[14] |
D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equations, Kinet. Relat. Models, 6 (2013), 989-1009.
doi: 10.3934/krm.2013.6.989. |
[15] |
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. |
[16] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. Henri Poincaré Anal. Non Linéare, 1 (1984), 109-145. |
[17] |
M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Commun. in Partial Differential Equations, 12 (1987), 1133-1173.
doi: 10.1080/03605308708820522. |
[18] |
E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Springer-Verlag, New York, 1995. |
[19] |
L. Zeng, Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type, J. Diff. Eqns., 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, Contemp. Math., 221 (1999), 1-29.
doi: 10.1090/conm/221/03116. |
[2] |
J. P. Albert, J. L. Bona and D. B. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, Phys. D, 24 (1987), 343-366.
doi: 10.1016/0167-2789(87)90084-4. |
[3] |
J. P. Albert, J. L. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. R. Soc. Lond., 453 (1997), 1233-1260.
doi: 10.1098/rspa.1997.0068. |
[4] |
J. Angulo Pava, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, American Mathematical Society, Providence, RI, 2009.
doi: 10.1090/surv/156. |
[5] |
H. Chen and J. L. Bona, Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Differential Equations, 3 (1998), 51-84. |
[6] |
R. R. Coifman and Y. Meyer, Au-delà des Opérateurs Pseudo-différentiels, Astérisque, 1978. |
[7] |
M. Ehrnström, J. Escher and L. Pei, A note on the local well-posedness for the Whitham equation, Elliptic and Parabolic Equations, Springer Proceedings in Mathematics & Statistics, 119 (2015), 63-75.
doi: 10.1007/978-3-319-12547-3_3. |
[8] |
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. |
[9] |
M. Ehrnström and H. Kalisch, Global bifurcation for the Whitham equation, Math. Model. Nat. Phenom., 8 (2013), 13-30.
doi: 10.1051/mmnp/20138502. |
[10] |
R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[11] |
B. Guo and D. Huang, Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys., 53 (2012), 083702, 15pp.
doi: 10.1063/1.4746806. |
[12] |
V. M. Hur and M. A. Johnson, Modulational instability in the Whitham equation for water waves, Studies in Applied Mathematics, 134 (2015), 120-143.
doi: 10.1111/sapm.12061. |
[13] |
C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Physica D, 295/296 (2015), 46-65.
doi: 10.1016/j.physd.2014.12.004. |
[14] |
D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equations, Kinet. Relat. Models, 6 (2013), 989-1009.
doi: 10.3934/krm.2013.6.989. |
[15] |
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. |
[16] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. Henri Poincaré Anal. Non Linéare, 1 (1984), 109-145. |
[17] |
M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Commun. in Partial Differential Equations, 12 (1987), 1133-1173.
doi: 10.1080/03605308708820522. |
[18] |
E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Springer-Verlag, New York, 1995. |
[19] |
L. Zeng, Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type, J. Diff. Eqns., 188 (2003), 1-32.
doi: 10.1016/S0022-0396(02)00061-X. |
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