Abstract
We prove existence and uniqueness of the solutions of Kolmogorov Petrovskii-Piskunov (KPP) equation. We study asymptotic stability and instability of the equilibrium solution u(x; t) = 0 of KPP equation with subject to the traveling wave solutions. We show that KPP equation has not got any periodic traveling wave solution. Also, we obtain some exact traveling wave solutions of KPP equation by the first integral method.
Highlights
In this paper, we are interested in the equation of Kolmogorov-Petrovskii-Piskunov ut uxx + u + u2 + u3 = 0; x 2 R; t 2 [0; 1) (1)with the initial condition u(0; x) = u0(x); x 2 R: (2)KPP equation ...rst appeared in the genetics model for the spread of an advantageous gene through a population [12]
KPP equation contains various well known nonlinear equations in mathematical physics; In the case of = 1; = 0; = 1; it reduces to the Newell-Whitehead equation, for = a; = (a + 1); = 1; it is called FitzHugh-Nagumo equation and for = 1; = 1; = 0; it is a special case of Fisher equation ut uxx = u u2: The reason for our interest in the KPP equation is that there exist solutions to the KPP equation whose qualitative behavior resembles the traveling wave solutions
The ...rst integral method has not been applied to Eq (1) for the same purpose
Summary
Existence and uniqueness of solutions, asymptotic stability, instability, periodicity, traveling wave solutions, ...rst integral method. Our aim is ...rstly to study the asymptotic stability and instability of zero solution of KPP equation with subject to all traveling wave solutions by means of qualitative theory of ordinary d¤erential equations, secondly to explore the periodic traveling wave solution of KPP equation and thirdly to ...nd some exact traveling wave solutions of KPP equation by using the ...rst integral method. In the ...nal section, we showed that if our conditions are satis...ed, a traveling wave solution that we obtained can approach to zero
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