American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity
  • CPAA Home
  • This Issue
  • Next Article
    Multiple solutions for a fractional nonlinear Schrödinger equation with local potential
November  2017, 16(6): 2125-2132. doi: 10.3934/cpaa.2017105

Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels

Albert Erkip 1,, and  Abba I. Ramadan 1,2,

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

* Corresponding author

Received  December 2016 Revised  February 2017 Published  July 2017

Fund Project: The second author is supported by TUBITAK Grant 2215 -Graduate Scholarship Programme for International Students.

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.

Keywords: olitary waves, bell-shaped functions, nonlocal wave equations, variational methods.
Mathematics Subject Classification: Primary: 74H20, 74J30; Secondary: 35Q51, 35A15.
Citation: Albert Erkip, Abba I. Ramadan. Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2125-2132. doi: 10.3934/cpaa.2017105
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418.   Google Scholar

[6]

E. H. Lieb and M. Loss, Analysis Volume 14, Graduate Studies in Mathematics, AMS, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418.   Google Scholar

[6]

E. H. Lieb and M. Loss, Analysis Volume 14, Graduate Studies in Mathematics, AMS, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[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.  Google Scholar

[1]

Riccardo Aragona, Marco Calderini, Roberto Civino, Massimiliano Sala, Ilaria Zappatore. Wave-shaped round functions and primitive groups. Advances in Mathematics of Communications, 2019, 13 (1) : 67-88. doi: 10.3934/amc.2019004
[2]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[3]

Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775
[4]

Mathias Nikolai Arnesen. Existence of solitary-wave solutions to nonlocal equations. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3483-3510. doi: 10.3934/dcds.2016.36.3483
[5]

Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1589-1601. doi: 10.3934/dcds.2010.28.1589
[6]

Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic & Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729
[7]

Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459
[8]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659
[9]

Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i
[10]

Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657
[11]

H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119
[12]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
[13]

F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier. Dispersive waves with multiple tunnel effect on a star-shaped network. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 783-791. doi: 10.3934/dcdss.2013.6.783
[14]

H. A. Erbay, S. Erbay, A. Erkip. The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6101-6116. doi: 10.3934/dcds.2016066
[15]

Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211
[16]

Peter Giesl, Sigurdur Hafstein. Review on computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2291-2331. doi: 10.3934/dcdsb.2015.20.2291
[17]

Farah Abou Shakra. Asymptotics of wave models for non star-shaped geometries. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 347-362. doi: 10.3934/dcdss.2014.7.347
[18]

Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331
[19]

Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048
[20]

Nar Rawal, Wenxian Shen, Aijun Zhang. Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1609-1640. doi: 10.3934/dcds.2015.35.1609

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (95)
  • HTML views (51)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences