I am interested in a flame propagation in 1D which is governed by a nonlinear convection-diffusion-reaction equation system:

$$ u_t+C(u)u_x+(D(u)\cdot u_x)_x + R(u) = 0,x\in (-\infty,+\infty) \\ u(-\infty,t) = u_- \\ u_x(\infty,t) = 0 $$

$u=[T,c_i,v]^T$.I see if I use $u=u(\theta)=u(x+ct)$. I can convert the original problem to $$ N(c)[u]=0\\ u(\theta=-\infty) = u_- \text{ unburned mixture}\\ u_x(\theta=\infty) = 0 \text{ burnt} $$

It seems like a nonlinear eigen value problem of an nonlinear operator $N(c)$ with one parameter $c>0$.

My problem is: what is the existence condition of a stable solitary wave-like solution to the problem? I am especially interested in the constraints on the reaction term and un-burned gas condition.

Further, what is the best numerical solution to this kind of problem?

In combustion theory, not all initial composition can generate a flame. The heat release rate should be high enough to maintain a combustion wave. But there is no clear conclusion when such a solution exist.

In soliton theory, mathematicians developed a theory named as `Lax-pair`

. But I cannot understand it in the sense of physics. It seems every wave speed is possible and no boundary condition is present in the theory.