I have the following quasi-linear ode (1)(adsbygoogle = window.adsbygoogle || []).push({});

[tex]-\frac{d^2 u}{dx^2}+\frac{\lambda}{(1+u)^2}=0[/tex]

with boundary conditions [itex]u(\pm \frac{1}{2})=0[/itex] and [itex]\lambda>0[/itex].

I've proven this equation to have two solutions for [itex]\lambda<\lambda^*[/itex], one for [itex]\lambda=\lambda^*[/itex] and none for [itex]\lambda*<\lambda[/itex].

Also, i know that [itex]-1< u(0) \le u(x) \le 0[/itex] and that [itex] u[/itex] is convex. I can plot the bifurcation diagram ([itex]\|u\|_\infty \quad vs \quad \lambda[/itex]) and find out that in [itex]\lambda=\lambda^*[/itex] I have a subcritical bifurcation.

With all this information, I need to find out wich branch of the bifurcation diagram is the stable one. In order to do that, I state the linealized eigenvalue problem (2) around a solution [itex]u_s(x)[/itex] of (1)

[tex]-\frac{d^2 u}{d x^2}-\left[\frac{2\lambda}{(1+u_s(x))^3}+\mu\right]u=0[/tex]

and here is where i get stuck.

The eigenvalue [itex]\mu=0[/itex] will tell me where i loose stability, so I need to solve (2) with [itex]\mu=0[/itex] right? And if I found a [itex]\lambda[/itex] where the solution is nontrivial then I'll know wich branch is the unstable one. Is that correct?

How can I calculate such [itex]\lambda[/itex]?

I've read that if I take the operator

[tex]A=-\frac{d^2}{d x^2}-\frac{2\lambda}{(1+u_s(x))^3}[/tex],

since

[tex]\frac{2\lambda}{\left[1+u_s(x)\right]^3} \le \frac{2\lambda}{\left[1+u_s(0)\right]^3}=c[/tex]

then i can define the operator

[tex]A_c=A+(c+1)[/tex]

wich is strongly positive and has eigenvalues [itex]\mu_0-c-1,\mu_1-c-1,...,\mu_n-c-1,...[/itex] where [itex]\mu_0,\mu_1,...,\mu_n,...[/itex] are the eigenvalues of (2). If I calculate the eigenvalues of [itex]A_s[/itex] then i can calculate [itex]\lambda[/itex]. Is this a better way or am I just complicating things?

Other apporach i did was to use the WKB method in (2) but i get stuck trying to apply boundary conditions in my integrals.

I need someone to help me clarify my mind and tell me if im getting somewhere or im just waisting everyones time...

sorry for bad english and thx for help

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# Stability for quasi-linear ODE

Can you offer guidance or do you also need help?

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