Browsing by Author "Yang, Chi-Ru"

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Diversity of traveling wave solutions in FitzHugh-Nagumo type equations
(2009-08-15)
Hsu, Cheng-Hsiung
;
Yang, Ting-Hui
;
Yang, Chi-Ru
;
應用數學系
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Department of Applied Mathematics
In this work we consider the diversity of traveling wave solutions of the FitzHugh-Nagumo type equations u(t) = u(xx) + f (u, w), w(t) = epsilon g(u, w), where f(u, w) = u(u - a(w))(1 - u) for some smooth function a(w) and g(u, w) = u - w. When a(w) crosses zero and one, the corresponding profile equation possesses special turning points which result in very rich dynamics. In [W. Liu, E. Van Vleck, Turning points and traveling waves in FitzHugh-Nagumo type equations. J. Differential Equations 225 (2006) 381-410], Liu and Van Vleck examined traveling waves whose slow orbits lie only on two portions of the slow manifold, and obtained the existence results by using the geometric singular perturbation theory. Based on the ideas of their work, we study the co-existence of different traveling waves whose slow orbits could involve all portions of the slow manifold. There are more complicated and richer dynamics of traveling waves than those of [W. Liu, E. Van Vleck, Turning points and traveling waves in FitzHugh-Nagumo type equations, J. Differential Equations 225 (2006) 381-410]. We give a complete classification of all different fronts of traveling waves, and provide an example to support our theoretical analysis. (C) 2009 Elsevier Inc. All rights reserved.
Existence of traveling wave solutions for diffusive predator-prey type systems
(2012-02-15)
Hsu, Cheng-Hsiung
;
Yang, Chi-Ru
;
Yang, Ting-Hui
;
Yang, Tzi-Sheng
;
應用數學系
In this work we investigate the existence of traveling wave solutions for a class of diffusive predator-prey type systems whose each nonlinear term can be separated as a product of suitable smooth functions satisfying some monotonic conditions. The profile equations for the above system can be reduced as a four-dimensional ODE system, and the traveling wave solutions which connect two different equilibria or the small amplitude traveling wave train solutions are equivalent to the heteroclinic orbits or small amplitude periodic solutions of the reduced system. Applying the methods of Wazewski Theorem, LaSalle's Invariance Principle and Hopf bifurcation theory, we obtain the existence results. Our results can apply to various kinds of ecological models. (C) 2011 Elsevier Inc. All rights reserved.
Smooth solutions of the one-dimensional compressible Euler equation with gravity
(2016-01-05)
Hsu, Cheng-Hsiung
;
Lin, Song-Sun
;
Yang, Chi-Ru
;
應用數學系
;
Department of Applied Mathematics
We study one-dimensional motions of polytropic gas governed by the compressible Euler equations. The problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions. But it is difficult to justify the existence of long-time true solutions for which this time periodic solution is the first approximation. The situation is in contrast to the problem of free motions without gravity. The reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum. Due to this reason, we try to find a family of solutions expanded by a small parameter and apply the Nash-Moser Theorem to justify this expansion. Note that the application of Nash-Moser Theorem is necessary for the sake of conquest of the trouble with loss of regularities, and the justification of the applicability requires a very delicate analysis of the problem. (C) 2015 Elsevier Inc. All rights reserved.