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Multiple solutions for a fractional nonlinear Schrödinger equation with local potential
Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels
| 1. | Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey |
| 2. | Department of Mathematics, 405 Snow Hall, University of Kansas, Lawrence, KS 66045 |
In this article we are concerned with the existence of traveling wave solutions of a general class of nonlocal wave equations: $~u_{tt}-a^{2}u_{xx}=(β * u^{p})_{xx}$, $~p>1$. Members of the class arise as mathematical models for the propagation of waves in a wide variety of situations. We assume that the kernel $β $ is a bell-shaped function satisfying some mild differentiability and growth conditions. Taking advantage of growth properties of bell-shaped functions, we give a simple proof for the existence of bell-shaped traveling wave solutions.
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Traveling waves for monomer chains with precompression, Nonlinearity, 26 (2013), 539-349.
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doi: 10.3934/dcdsb.2004.4.1065. |
show all references
References:
| [1] |
C. Babaoglu, H. A. Erbay and A. Erkip,
Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations, Nonlinear Anal., 77 (2013), 82-93.
doi: 10.1016/j.na.2012.09.001. |
| [2] |
N. Duruk, H. A. Erbay and A. Erkip,
Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, 23 (2010), 107-118.
doi: 10.1088/0951-7715/23/1/006. |
| [3] |
J. M. English and R. L. Pego,
On the solitarywave pulse in a chain of beads, Proc. AMS, 133, 23 (2005), 1763-1768.
doi: 10.1090/S0002-9939-05-07851-2. |
| [4] |
H. A. Erbay, S. Erbay and A. Erkip,
Existence and stability of traveling waves for a class of nonlocal nonlinear equations, J. Mathematical Analysis and Applications, 425 (2015), 307-336.
doi: 10.1016/j.jmaa.2014.12.039. |
| [5] |
G. Friesecke and J. A. D. Wattis,
Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418.
|
| [6] |
E. H. Lieb and M. Loss, Analysis Volume 14, Graduate Studies in Mathematics, AMS, 2001.
doi: 10.1090/gsm/014. |
| [7] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. Henri Poincaré, 1 (1984), 109-145.
|
| [8] |
A. Stefanov and P. Kevrekidis,
On the existence of solitary traveling waves for generalized Hertzian chains, Journal of Nonlinear Science, 22 (2012), 327-349.
doi: 10.1007/s00332-011-9119-9. |
| [9] |
A. Stefanov and P. Kevrekidis,
Traveling waves for monomer chains with precompression, Nonlinearity, 26 (2013), 539-349.
doi: 10.1088/0951-7715/26/2/539. |
| [10] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
| [11] |
C. Wolf,
A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems-B, 4 (2004), 1065-1089.
doi: 10.3934/dcdsb.2004.4.1065. |
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