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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.

MSC:
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)
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