zbMATH — the first resource for mathematics

Pulse dynamics in a three-component system: Existence analysis. (English) Zbl 1173.35068
The authors of this very interesting paper study a system of three partial differential equations. This system applicable to chemistry and biology consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor. The system has the form,
\[ \begin{aligned} U_t&= D_UU_{xx}+f(U)-k_3V-k_4W+k_1,\\ \tau V_t&= D_VV_{xx}+U-V,\\ \theta W_t&= D_WW_{xx}+U-W, \end{aligned} \] where \(U,V,W\) are real-valued functions of \(t\in \mathbb R ^+\) and \(x\in \mathbb R \), and the subscripts indicate partial derivatives, \(\tau \) and \(\eta \) are positive parameters. The existence of stationary one-pulse and two-pulse solutions and travelling one-pulse solutions on the real line are demonstrated. It turns out that the parameter regimes influence on the solutions therefore the bifurcation theory for one- and two-pulse solutions as well traveling one-pulse solutions could be applied. An interesting result obtained here is that there exists the bifurcation from a stationary to a traveling pulse. For two pulse solutions the third component is essential since the reduced bistable two-component system does not support them. The second interesting result is the existence of saddle-node bifurcation in which two-pulse solutions are created. By numerical examples are shown stable breathing one- and two-pulse solutions, scattering of two pulses as well spatio-temporal dynamics of a solution with symmetric four-pulse initial data.

35K55 Nonlinear parabolic equations
35B32 Bifurcations in context of PDEs
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35K45 Initial value problems for second-order parabolic systems
35B25 Singular perturbations in context of PDEs
92D25 Population dynamics (general)
Full Text: DOI
[1] Blom J.G., Zegeling P.A.: Algorithm 731: a moving-grid interface for systems of one-dimensional time-dependent partial differential equations. ACM Trans. Math. Softw. 20, 194–214 (1994) · Zbl 0889.65099 · doi:10.1145/178365.178391
[2] Bode M., Liehr A.W., Schenk C.P., Purwins H.-G.: Interaction of dissipative solitons: particle-like behavior of localized structures in a three-component reaction-diffusion system. Physica D 161, 45–66 (2002) · Zbl 0985.35096 · doi:10.1016/S0167-2789(01)00360-8
[3] Doelman A., Gardner R.A., Kaper T.J.: Large stable pulse solutions in reaction-diffusion equations. Indian Univ. Math. J. 50(1), 443–507 (2001) · Zbl 0994.35058 · doi:10.1512/iumj.2001.50.1873
[4] Doelman A., Kaper T.J., van der Ploeg H.: Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation. Meth. Appl. Anal. 8(3), 387–414 (2001) · Zbl 1006.92008
[5] Doelman, A., Gardner, R.A., Kaper, T.J.: A stability index analysis of 1-D patterns of the Gray–Scott model. Mem. AMS 155(737) (2002) · Zbl 0994.35059
[6] Doelman A., Iron D., Nishiura Y.: Destabilization of fronts in a class of bi-stable systems. SIAM Math. J. Anal. 35(6), 1420–1450 (2004) · Zbl 1063.35023 · doi:10.1137/S0036141002419242
[7] Evans J.W., Fenichel N., Feroe J.A.: Double impulse solutions in nerve axon equations. SIAM J. Appl. Math. 42, 219–234 (1982) · Zbl 0512.92006 · doi:10.1137/0142016
[8] Fenichel N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971) · Zbl 0246.58015 · doi:10.1512/iumj.1971.21.21017
[9] Fenichel N.: Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations 31, 53–98 (1979) · Zbl 0476.34034 · doi:10.1016/0022-0396(79)90152-9
[10] Hastings S.: Single and multiple pulse waves for the FitzHugh–Nagumo equations. SIAM J. Appl. Math. 42, 247–260 (1982) · Zbl 0503.92009 · doi:10.1137/0142018
[11] Jones, C.K.R.T.: Geometric singular perturbation theory. In: Johnson, R. (ed.) Dynamical Systems, Montecatini Terme, 1994: Lecture Notes in Mathematics, vol. 1609. Springer-Verlag (1995)
[12] Jones C.K.R.T., Kopell N.: Tracking invariant manifolds with differential forms in singularly perturbed systems. J. Differential Equations 108(1), 64–88 (1994) · Zbl 0796.34038 · doi:10.1006/jdeq.1994.1025
[13] Jones C.K.R.T., Kaper T.J., Kopell N.: Tracking invariant manifolds up to exponentially small errors. SIAM J. Math. Anal. 27(2), 558–577 (1996) · Zbl 0871.58072 · doi:10.1137/S003614109325966X
[14] Kaper T.J.: An introduction to geometric methods and dynamical systems theory for singular perturbation problems. Proc. Sympos. Appl. Math. 56, 85–131 (1999)
[15] Nishiura Y., Teramoto T., Ueda K.-I.: Scattering and separators in dissipative systems. Phys. Rev. E 67, 056210 (2003) · doi:10.1103/PhysRevE.67.056210
[16] Nishiura Y., Teramoto T., Ueda K.-I.: Scattering of traveling spots in dissipative systems. CHAOS 15(4), 047509 (2005) · Zbl 1144.37393 · doi:10.1063/1.2087127
[17] Nishiura Y., Teramoto T., Yuan X., Ueda K.-I.: Dynamics of traveling pulses in heterogeneous media. CHAOS 17(3), 031704 (2007) · Zbl 1163.37356 · doi:10.1063/1.2778553
[18] Or-Guil M., Bode M., Schenk C.P., Purwins H.-G.: Spot bifurcations in three-component reaction-diffusion systems: the onset of propagation. Phys. Rev. E 57, 6432–6437 (1998) · doi:10.1103/PhysRevE.57.6432
[19] Rasker, A.P.: Pulses in a bi-stable reaction-diffusion system. MA Thesis, KdV Inst., Univ. Amsterdam, The Netherlands (2005)
[20] Robinson C.: Sustained resonance for a nonlinear system with slowly-varying coefficients. SIAM J. Math. Anal. 14, 847–860 (1983) · Zbl 0523.34035 · doi:10.1137/0514066
[21] Rubin J., Jones C.K.R.T.: Existence of standing pulse solutions to an inhomogeneous reaction-diffusion system. J. Dynam. Differential Equations 10, 1–35 (1998) · Zbl 0896.34041 · doi:10.1023/A:1022651311294
[22] Schenk C.P., Or-Guil M., Bode M., Purwins H.-G.: Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains. PRL 78(19), 3781–3784 (1997) · doi:10.1103/PhysRevLett.78.3781
[23] van Heijster, P., Doelman, A., Kaper, T.J.: Pulse dynamics in a three-component system: stability and bifurcations. To appear in Physica D (2008) · Zbl 1153.37437
[24] Yang L., Zhabotinsky A.M., Epstein I.R.: Jumping solitary waves in an autonomous reaction-diffusion system with subcritical wave instability. Phys. Chem. Chem. Phys. 8, 4647–4651 (2006) · doi:10.1039/b609214d
[25] Yuan X., Teramoto T., Nishiura Y.: Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system. Phys. Rev. E 75, 036220 (2007) · doi:10.1103/PhysRevE.75.036220
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.