• Previous Article
    Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity
  • DCDS-B Home
  • This Issue
  • Next Article
    Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions
June  2017, 22(4): 1601-1633. doi: 10.3934/dcdsb.2017033

Stability of traveling waves for autocatalytic reaction systems with strong decay

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

* Corresponding author

Received  May 2016 Revised  June 2016 Published  November 2016

This paper is concerned with the spatial decay and stability of travelling wave solutions for a reaction diffusion system $u_{t}=d u_{xx}-uv$, $v_{t}=v_{xx}+uv-Kv^{q}$ with $q>1$. By applying Centre Manifold Theorem and detailed asymptotic analysis, we get the precise spatial decaying rate of the travelling waves with noncritical speeds. Further by applying spectral analysis, Evans function method and some numerical simulation, we proved the spectral stability and the linear exponential stability of the waves with noncritical speeds in some weighted spaces.

Citation: Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1601-1633. doi: 10.3934/dcdsb.2017033
References:
[1]

A. L. Afendikov and T. J. Bridges, Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 257-272.  doi: 10.1098/rspa.2000.0665.  Google Scholar

[2]

J. AlexanderR. Gardner and C. Jones, A topological invariant arising in the stability of travelling waves, J. Reine Angew. Math., 410 (1990), 167-212.   Google Scholar

[3]

J. C. Alexander and R. Sachs, Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation, Nonlinear World, 2 (1995), 471-507.   Google Scholar

[4]

L. Allen and T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numer. Math., 92 (2002), 197-232.  doi: 10.1007/s002110100365.  Google Scholar

[5]

B. Barker, J. Humpherys and K. Zumbrun, STABLAB: A MATLAB-based Numerical Library for Evans Function Computation Version 1. 0 preprint. Available from: http://impact.byu.edu/stablab/STABLAB_1.0_doc.pdf. Google Scholar

[6]

J. Billingham and D. J. Needham, A note on the properties of a family of traveling wave solutions arising in cubic autocatalysis, Dynam. Stability Systems, 6 (1991), 33-49.  doi: 10.1080/02681119108806105.  Google Scholar

[7]

T. J. BridgesG. Derks and G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Phys. D, 172 (2002), 190-216.  doi: 10.1016/S0167-2789(02)00655-3.  Google Scholar

[8]

J. Carr, Application Of Centre Manifold Theory Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[9]

X. ChenY. Qi and Y. Zhang, Existence of traveling waves of auto-catalytic systems with decay, J. Differential Equations, 260 (2016), 7982-7999.  doi: 10.1016/j.jde.2016.02.009.  Google Scholar

[10]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations D. C. Heath and Co. , Boston, Mass, 1965.  Google Scholar

[11]

S. Focant and Th. Galley, Existence and stability of propagating fronts for an autocatalytic reactio-diffuion systems, Physica D, 120 (1998), 346-368.  doi: 10.1016/S0167-2789(98)00096-7.  Google Scholar

[12]

S.-C. Fu and J.-C. Tsai, The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay, J. Differential Equations, 256 (2014), 3335-3364.  doi: 10.1016/j.jde.2014.02.009.  Google Scholar

[13]

P. GrayJ. F. Griffiths and S. K. Scott, Oscillations, Glow and Ignition in Carbon Monoxide Oxidation in an Open System Ⅰ. Experimental Studies of the Ignition Diagram and the Effects of Added Hydrogen, Proc. R. Soc. Lond. A, 397 (1985), 21-44.  doi: 10.1098/rspa.1985.0002.  Google Scholar

[14]

P. Gray, Instabilities and oscillations in chemical reactions in closed and open systems, Proc. R. Soc. Lond. A, 415 (1988), 1-34.   Google Scholar

[15]

A. HannaA. Saul and K. Showalter, Detailed studies of propagating fronts in the iodate oxidation of arsenous acid, J. Am. Chem. Soc., 104 (1982), 3838-3844.  doi: 10.1021/ja00378a011.  Google Scholar

[16]

D. Henry, Geometric Theory Of Semilinear Parabolic Equations Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[17]

W. Huang, Uniqueness of traveling wave solutions for a biological reaction-diffusion equation, J. Math. Anal. Appl., 316 (2006), 42-59.  doi: 10.1016/j.jmaa.2005.04.084.  Google Scholar

[18]

P. Howard and K. Zumbrun, The Evans functions and stbility criteria for degenerate viscous shcok waves, Discrete Contin. Dyn. Syst., 10 (2004), 837-855.  doi: 10.3934/dcds.2004.10.837.  Google Scholar

[19]

E. Jakab, D. Horvath, J. H. Merkin, S. K. Scott, P. L. Simon and A. Toth, Isothermal flame balls: Effect of autocatalyst decay Phys. Rev. E 66 (2002), 016207. doi: 10.1103/PhysRevE.66.016207.  Google Scholar

[20]

Y. Li and Y. Wu, Stability of traveling waves with non-critical speeds for double degenerate Fisher-type equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 149-170.  doi: 10.3934/dcdsb.2008.10.149.  Google Scholar

[21]

Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521.  doi: 10.1137/100814974.  Google Scholar

[22]

Y. Li and Y. Wu, Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems, Math. Model. Nat. Phenom., 8 (2013), 104-132.  doi: 10.1051/mmnp/20138308.  Google Scholar

[23]

P. M. McCabeJ. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay.I. Permanent form travelling waves, SIAM J. Appl. Math., 59 (1999), 870-899.  doi: 10.1137/S0036139996312594.  Google Scholar

[24]

P. M. McCabeJ. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay.Ⅱ. The initial boundary value problem, SIAM J. Appl. Math., 60 (2000), 1707-1748.  doi: 10.1137/S0036139998344775.  Google Scholar

[25]

J. H. Merkin and D. J. Needham, The development of travelling waves in a simple isothermal chemical system.Ⅳ. Quadratic autocatalysis with quadratic decay, Proc. R. Soc. Lond. A, 434 (1991), 531-554.   Google Scholar

[26]

J. H. MerkinD. J. Needham and S. K. Scott, A simple model for sustained oscillations in isothermal branch-chain or autocatalytic reactions in a well stirred open system. Ⅰ. Stationary states and local stabilities, Proc. Roy. Soc. London Ser. A, 398 (1985), 81-100.  doi: 10.1098/rspa.1985.0026.  Google Scholar

[27]

J. H. MerkinD. J. Needham and S. K. Scott, The development of travelling waves in a simple isothermal chemical system.I. Quadratic autocatalysis with linear decay, Proc. Roy. Soc. London Ser. A, 424 (1989), 187-209.  doi: 10.1098/rspa.1989.0075.  Google Scholar

[28]

D. J. Needham, A note on the global asymptotic stability of the unreacting state in a simple model for quadratic autocatalysis with linear decay, Z. Angew. Math. Phys., 42 (1991), 455-459.  doi: 10.1007/BF00945715.  Google Scholar

[29]

D. J. Needham and J. H. Merkin, The development of travelling waves in a simple isothermal chemical system with general orders of autocatalysis and decay, Philos. Trans. Roy. Soc. London Ser. A, 337 (1991), 261-274.  doi: 10.1098/rsta.1991.0122.  Google Scholar

[30]

B. S. Ng and W. H. Reid, The compound matrix method for ordinary differential systems, J. Comput. Phys., 58 (1985), 209-228.   Google Scholar

[31]

A. Pazy, Semigroups Of Linear Operators And Applications To Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[32]

B. P. Palka, An Introduction To Complex Function Theory Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0975-1.  Google Scholar

[33]

R. L. PegoP. Smereka and M. I. Weinstein, Oscillatory instability of traveling waves for a KdV-Burgers equation, Phys. D, 67 (1993), 45-65.  doi: 10.1016/0167-2789(93)90197-9.  Google Scholar

[34]

R. L. Pego and M. I. Weinstein, Eigenvalues, and instability of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47-94.  doi: 10.1098/rsta.1992.0055.  Google Scholar

[35]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes In C: The Art Of Scientific Computing. Second Edition Cambridge University Press, Cambridge, 1992.  Google Scholar

[36]

L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs With MATLAB Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511615542.  Google Scholar

[37]

Y. WuX. Xing and Q. Ye, Stability of travelling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 16 (2006), 47-66.  doi: 10.3934/dcds.2006.16.47.  Google Scholar

[38]

Q. Ye, Z. Y. Li, M. X. Wang and Y. Wu, Introduction To Reaction Diffusion Equation. Second Edition Science Press, Beijing, 2011. Google Scholar

show all references

References:
[1]

A. L. Afendikov and T. J. Bridges, Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 257-272.  doi: 10.1098/rspa.2000.0665.  Google Scholar

[2]

J. AlexanderR. Gardner and C. Jones, A topological invariant arising in the stability of travelling waves, J. Reine Angew. Math., 410 (1990), 167-212.   Google Scholar

[3]

J. C. Alexander and R. Sachs, Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation, Nonlinear World, 2 (1995), 471-507.   Google Scholar

[4]

L. Allen and T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numer. Math., 92 (2002), 197-232.  doi: 10.1007/s002110100365.  Google Scholar

[5]

B. Barker, J. Humpherys and K. Zumbrun, STABLAB: A MATLAB-based Numerical Library for Evans Function Computation Version 1. 0 preprint. Available from: http://impact.byu.edu/stablab/STABLAB_1.0_doc.pdf. Google Scholar

[6]

J. Billingham and D. J. Needham, A note on the properties of a family of traveling wave solutions arising in cubic autocatalysis, Dynam. Stability Systems, 6 (1991), 33-49.  doi: 10.1080/02681119108806105.  Google Scholar

[7]

T. J. BridgesG. Derks and G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Phys. D, 172 (2002), 190-216.  doi: 10.1016/S0167-2789(02)00655-3.  Google Scholar

[8]

J. Carr, Application Of Centre Manifold Theory Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[9]

X. ChenY. Qi and Y. Zhang, Existence of traveling waves of auto-catalytic systems with decay, J. Differential Equations, 260 (2016), 7982-7999.  doi: 10.1016/j.jde.2016.02.009.  Google Scholar

[10]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations D. C. Heath and Co. , Boston, Mass, 1965.  Google Scholar

[11]

S. Focant and Th. Galley, Existence and stability of propagating fronts for an autocatalytic reactio-diffuion systems, Physica D, 120 (1998), 346-368.  doi: 10.1016/S0167-2789(98)00096-7.  Google Scholar

[12]

S.-C. Fu and J.-C. Tsai, The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay, J. Differential Equations, 256 (2014), 3335-3364.  doi: 10.1016/j.jde.2014.02.009.  Google Scholar

[13]

P. GrayJ. F. Griffiths and S. K. Scott, Oscillations, Glow and Ignition in Carbon Monoxide Oxidation in an Open System Ⅰ. Experimental Studies of the Ignition Diagram and the Effects of Added Hydrogen, Proc. R. Soc. Lond. A, 397 (1985), 21-44.  doi: 10.1098/rspa.1985.0002.  Google Scholar

[14]

P. Gray, Instabilities and oscillations in chemical reactions in closed and open systems, Proc. R. Soc. Lond. A, 415 (1988), 1-34.   Google Scholar

[15]

A. HannaA. Saul and K. Showalter, Detailed studies of propagating fronts in the iodate oxidation of arsenous acid, J. Am. Chem. Soc., 104 (1982), 3838-3844.  doi: 10.1021/ja00378a011.  Google Scholar

[16]

D. Henry, Geometric Theory Of Semilinear Parabolic Equations Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[17]

W. Huang, Uniqueness of traveling wave solutions for a biological reaction-diffusion equation, J. Math. Anal. Appl., 316 (2006), 42-59.  doi: 10.1016/j.jmaa.2005.04.084.  Google Scholar

[18]

P. Howard and K. Zumbrun, The Evans functions and stbility criteria for degenerate viscous shcok waves, Discrete Contin. Dyn. Syst., 10 (2004), 837-855.  doi: 10.3934/dcds.2004.10.837.  Google Scholar

[19]

E. Jakab, D. Horvath, J. H. Merkin, S. K. Scott, P. L. Simon and A. Toth, Isothermal flame balls: Effect of autocatalyst decay Phys. Rev. E 66 (2002), 016207. doi: 10.1103/PhysRevE.66.016207.  Google Scholar

[20]

Y. Li and Y. Wu, Stability of traveling waves with non-critical speeds for double degenerate Fisher-type equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 149-170.  doi: 10.3934/dcdsb.2008.10.149.  Google Scholar

[21]

Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521.  doi: 10.1137/100814974.  Google Scholar

[22]

Y. Li and Y. Wu, Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems, Math. Model. Nat. Phenom., 8 (2013), 104-132.  doi: 10.1051/mmnp/20138308.  Google Scholar

[23]

P. M. McCabeJ. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay.I. Permanent form travelling waves, SIAM J. Appl. Math., 59 (1999), 870-899.  doi: 10.1137/S0036139996312594.  Google Scholar

[24]

P. M. McCabeJ. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay.Ⅱ. The initial boundary value problem, SIAM J. Appl. Math., 60 (2000), 1707-1748.  doi: 10.1137/S0036139998344775.  Google Scholar

[25]

J. H. Merkin and D. J. Needham, The development of travelling waves in a simple isothermal chemical system.Ⅳ. Quadratic autocatalysis with quadratic decay, Proc. R. Soc. Lond. A, 434 (1991), 531-554.   Google Scholar

[26]

J. H. MerkinD. J. Needham and S. K. Scott, A simple model for sustained oscillations in isothermal branch-chain or autocatalytic reactions in a well stirred open system. Ⅰ. Stationary states and local stabilities, Proc. Roy. Soc. London Ser. A, 398 (1985), 81-100.  doi: 10.1098/rspa.1985.0026.  Google Scholar

[27]

J. H. MerkinD. J. Needham and S. K. Scott, The development of travelling waves in a simple isothermal chemical system.I. Quadratic autocatalysis with linear decay, Proc. Roy. Soc. London Ser. A, 424 (1989), 187-209.  doi: 10.1098/rspa.1989.0075.  Google Scholar

[28]

D. J. Needham, A note on the global asymptotic stability of the unreacting state in a simple model for quadratic autocatalysis with linear decay, Z. Angew. Math. Phys., 42 (1991), 455-459.  doi: 10.1007/BF00945715.  Google Scholar

[29]

D. J. Needham and J. H. Merkin, The development of travelling waves in a simple isothermal chemical system with general orders of autocatalysis and decay, Philos. Trans. Roy. Soc. London Ser. A, 337 (1991), 261-274.  doi: 10.1098/rsta.1991.0122.  Google Scholar

[30]

B. S. Ng and W. H. Reid, The compound matrix method for ordinary differential systems, J. Comput. Phys., 58 (1985), 209-228.   Google Scholar

[31]

A. Pazy, Semigroups Of Linear Operators And Applications To Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[32]

B. P. Palka, An Introduction To Complex Function Theory Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0975-1.  Google Scholar

[33]

R. L. PegoP. Smereka and M. I. Weinstein, Oscillatory instability of traveling waves for a KdV-Burgers equation, Phys. D, 67 (1993), 45-65.  doi: 10.1016/0167-2789(93)90197-9.  Google Scholar

[34]

R. L. Pego and M. I. Weinstein, Eigenvalues, and instability of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47-94.  doi: 10.1098/rsta.1992.0055.  Google Scholar

[35]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes In C: The Art Of Scientific Computing. Second Edition Cambridge University Press, Cambridge, 1992.  Google Scholar

[36]

L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs With MATLAB Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511615542.  Google Scholar

[37]

Y. WuX. Xing and Q. Ye, Stability of travelling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 16 (2006), 47-66.  doi: 10.3934/dcds.2006.16.47.  Google Scholar

[38]

Q. Ye, Z. Y. Li, M. X. Wang and Y. Wu, Introduction To Reaction Diffusion Equation. Second Edition Science Press, Beijing, 2011. Google Scholar

Figure 1.  Wave profiles by using shooting method for $K=1.5$, $q=2$, $c=3>c^*$ (a) $d=1$, $u^*=1$, (b) $d=3$, $u^*=2.1853$
Figure 2.  Wave profiles for $d\!\!=\!\!1$, $u^*\!\!=\!\!1$, $c\!\!=\!\!3\!\!>\!\!c^*$, $K\!=\!1.5$, $q\!\!=\!\!2$, $sl\!\!=\!\!-500$, $sr\!\!=\!\!50$ and (a) $\xi_0\!\!=\!\!0$; (b) $\xi_0\!\!=\!\!-80$
Figure 3.  Wave profiles for $K\!\!=\!\!1.5$, $d\!\!=\!\!1$, $u^*\!\!=\!\!1$, $q\!=\!2$, $sr\!\!=\!\!50$, $\xi_0\!=\!0$ and (a) $sl\!\!=\!\!-500$ with different $c\!\!>\!\!c^*$; (b) $c\!\!=\!\!4$ with different starting point $sl$
Figure 4.  (a) The selected curve Γ for K=1:5, q=2, u=1, c=3 and d=1. (b) The numerical curves of E(Γ) for q = 1:5, K = 1:5 and d = 1 with different c > c
Figure 5.  The numerical curves of $E(\Gamma)$ for $q\!=\!2$ and $K\!\!>\!\!1$: (a) $K\!\!=\!\!1.5$, $c\!\!=\!\!3$ with different $d\!\in\!(0,d^*)$, (b) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$ with different $c\!\!>\!\!c^*$
Figure 6.  The numerical curves of $E(\Gamma)$ for $q\!=\!2$ and $0\!\!<\!\!K\!\!\leq\!\!1$: (a) $K\!\!=\!\!0.5$, $c\!\!=\!\!3$ with different $d\!\in\!(0,d^*)$, (b) $d\!\!=\!\!1$, $K\!\!=\!\!0.5$ with different $c\!\!>\!\!c^*$
Figure 7.  The numerical curves of $E(\Gamma)$ for $1\!<q\!<2$: (a) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$, $c\!\!=\!\!4$ with different $q\!\!\in\!\!(1,2)$, (b) $q\!\!=\!\!1.5$, $K\!=\!1.5$, $c\!\!=\!\!4$ with different $d\!\in\!(0,d^*)$
Figure 8.  The numerical curves of $E(\Gamma)$ for $q=2.5$: (a) $K\!\!=\!\!1.5$, $c\!\!=\!\!4$ with different $d\!\in\!(0,d^*)$ (b) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$ with different $c\!\!>\!\!c^*$
Table 1.  The values of $E(0)$ and $E(10^4)$ for $q=2$, $K>1$ with the parameters corresponding to the curves in Fig. 5, we just retain integers here, which is verified the estimates in Lemma 4.5
$K\!=\!1.5$, $c\!=\!3$$c\!=\!3$, $d\!=\!1$$K\!=\!1.5$, $d\!=\!1$
$d\!=\!0.7$ $d\!=\!1$ $d\!=\!3$$K\!=\!5$ $K\!=\!3$ $K\!=\!1.5$ $c\!=\!3$ $c\!=\!5$ $c\!=\!10$
$E(0)$109 77 23177 92 6878 66 136
$E(10^4)$49254 41166 2367241633 41262 4109341166 40480 40733
$K\!=\!1.5$, $c\!=\!3$$c\!=\!3$, $d\!=\!1$$K\!=\!1.5$, $d\!=\!1$
$d\!=\!0.7$ $d\!=\!1$ $d\!=\!3$$K\!=\!5$ $K\!=\!3$ $K\!=\!1.5$ $c\!=\!3$ $c\!=\!5$ $c\!=\!10$
$E(0)$109 77 23177 92 6878 66 136
$E(10^4)$49254 41166 2367241633 41262 4109341166 40480 40733
[1]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885

[2]

Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837

[3]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359

[4]

Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic & Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261

[5]

Aslihan Demirkaya, Milena Stanislavova. Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 197-209. doi: 10.3934/dcdsb.2018097

[6]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857

[7]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939

[8]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141

[9]

Judith R. Miller, Huihui Zeng. Multidimensional stability of planar traveling waves for an integrodifference model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 741-751. doi: 10.3934/dcdsb.2013.18.741

[10]

Massimiliano Guzzo, Giancarlo Benettin. A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 1-28. doi: 10.3934/dcdsb.2001.1.1

[11]

Salvador Cruz-García, Catherine García-Reimbert. On the spectral stability of standing waves of the one-dimensional $M^5$-model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1079-1099. doi: 10.3934/dcdsb.2016.21.1079

[12]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941

[13]

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

[14]

Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258

[15]

Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601

[16]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993

[17]

Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895

[18]

Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047

[19]

Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1

[20]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (15)
  • HTML views (3)
  • Cited by (0)

Other articles
by authors

[Back to Top]