# On the intermediate asymptotic solutions in some models of the combustion theory.

We consider the travelling wave solutions of a nonlinear parabolic equation of the

second order, namely the equation of the Kolmogorov — Petrovsky — Piskunov type with the heat

release function on the right–hand side being analytical. We found a new analytic representation

for such a solution or, more accurately, for its inverse function which is represented as a sum

of an explicitly calculated summand and an auxiliary function deﬁned on the unit interval. An

algorithm for calculating the Taylor coeﬃcients of that function at the right endpoint and at the

interior points of the interval is constructed.

We establish a suﬃcient condition for for the mentioned auxiliary function to be analytical

on the entire unit interval including its both endpoints. The obtained criterion for the

analyticity allowed us to distinguish a countable dense set of values among the spectrum of the

permissible values for the traveling wave velocity (the spectrum being a numerical ray deﬁned

by A.Kolmogorov, I.Petrovskii and N.Piskunov) for which the auxiliary function is analytic and

consequently the inverse of the traveling wave solution is approximately representable by an

explicit formula up to a term uniformly bounded on the unit interval.

There is a result of the analytical theory of the Abel deﬀerential equation. In the proof of the

criterion of analyticity we use a kind of Painleve test (or Fuchs – Kovalevskaya – Painleve test)

applied to an accessorial equation namely to the Abel equation of the second kind. It became

apparent that this equation satisﬁes the Painleve test when some additional parameter (deﬁned

in the text) takes the prescribed values. Moreover the family of solutions passed through the

corresponding singular point of the equation consist of analytical functions when the conditions

of test gets satisﬁed.

In the second part of the paper an analytic–numerical method is developed based on the

representation described above. The method is applied to the problem of intermediate asymptotic

regimes of the thermal combustion of a gas mixture reacting at the initial temperature under

the condition of similarity of concentration and temperature ﬁelds. Some numerical results of the

constructed method are presented.

**Keywords:** travelling wave solutions, ﬂame propagation, intermediate asymptotics, Kolmogorov –

Petrovskii – Piskunov equation, Abel equation of the second kind, Painleve test.