Search

We consider the hyperbolic generalization of Burgers equation with polynomial source term. The transformation of auto-B\"{a}cklund type was found. Application of the results is shown in the examples, where the pair of two stationary solutions produces kink and bi-kink solutions., Comment: 10 pages, 3 figures

Following the paper of Shimizu et al. (Phys Lett A 375:1566, 2011), we consider the Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using the geometric singular perturbation theory. The considered model can be re-written as a four-dimensional (locally three-dimensional) autonomous system, which under certain conditions has a folded saddle-node singularity and additionally can be treated as a three time scale one. Author is very grateful to prof. Peter De Maesschalck for many stimulating discussions. The presented research was supported by the Research Foundation Flanders (FWO) under grant number G.0939.10N and by Polish Ministry of Science and Higher Education.

We consider an extended three-dimensional Bonhoeffer-van der Pol oscillator which generalises the planar FitzHugh-Nagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. (Phys Lett A 374(36):3745-3751, 2010) and by Freire and Gallas (Phys Lett A 375:1097-1103, 2011). Focussing on a parameter regime which has hitherto been neglected, and in which the governing equations evolve on three distinct time-scales, we propose a reduction to a model problem that was formulated by Krupa et al. (J Appl Dyn Syst 7(2):361-420, 2008) as a canonical form for such systems. Based on results previously obtained in Krupa et al. (2008), we characterise completely the mixed-mode dynamics of the resulting three time-scale extended Bonhoeffer-van der Pol oscillator from the point of view of geometric singular perturbation theory, thus complementing the findings reported in Sekikawa et al. (2010). In particular, we specify in detail the mixed-mode patterns that are observed upon variation of a bifurcation parameter which is naturally obtained by combining two of the original parameters in the system, and we derive asymptotic estimates for the corresponding parameter intervals. We thereby also disprove a conjecture of Tu (SIAM J Appl Math 49(2): 331-343, 1989), where it was postulated that no stable periodic orbits of mixed-mode type can be observed in an equivalent extension of the Bonhoeffer-van der Pol equations.

It is well known that various wave patterns observed in open dissipative systems are described by nonlinear PDEs, being not, as a rule, completely integrable. Yet the information about the existence of solutions describing the wave patterns (periodic, kink-like, soliton-like regimes and so on) within the dissipative model can be obtained by means of qualitative theory methods. In this work we show how it is possible, using the self-similar reduction and the qualitative analysis, to find approximated solutions to evolutionary PDEs, describing the solitary wave regimes. We apply this approach to the nonlinear d'Alembert equation and the hyperbolic generalization of the Burgers equation.

Direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some spatially nonlocal hydrodynamic-type model. Special attention is paid to the construction of the kink-like and soliton-like solutions.

We analyze the conditions, which guarantee the existence of periodic and soliton-like traveling wave solutions in the non-local hydrodynamic model of structured media., 20 pages, 2 figures

We present a review of our recent works directed towards discovery of a periodic, kink-like and soliton-like travelling wave solutions within the models of transport phenomena and the mathematical biology. Analytical description of these wave patterns is carried out by means of our modification of the direct algebraic balance method. In the case when the analytical description fails, we propose to approximate invariant travelling wave solutions by means of an infinite series of exponential functions. The effectiveness of the method of approximation is demonstrated on a hyperbolic modification of Burgers equation., Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

We present new periodic, kink-like and soliton-like travelling wave solutions to the hyperbolic generalization of Burgers equation. To obtain them, we employ the classical and generalized symmetry methods and the ansatz-based approach, Comment: 12 pages, 8 figures