Which Branch of the Bifurcation Diagram is Stable?

[ReyChiquito]

I have the following quasi-linear ode (1)

$$-\frac{d^2 u}{dx^2}+\frac{\lambda}{(1+u)^2}=0$$

with boundary conditions $u(\pm \frac{1}{2})=0$ and $\lambda>0$.

I've proven this equation to have two solutions for $\lambda<\lambda^*$, one for $\lambda=\lambda^*$ and none for $\lambda*<\lambda$.
Also, i know that $-1< u(0) \le u(x) \le 0$ and that $u$ is convex. I can plot the bifurcation diagram ($\|u\|_\infty \quad vs \quad \lambda$) and find out that in $\lambda=\lambda^*$ I have a subcritical bifurcation.

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

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

and here is where i get stuck.

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

How can I calculate such $\lambda$?

I've read that if I take the operator

$$A=-\frac{d^2}{d x^2}-\frac{2\lambda}{(1+u_s(x))^3}$$,

since

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

then i can define the operator

$$A_c=A+(c+1)$$

wich is strongly positive and has eigenvalues $\mu_0-c-1,\mu_1-c-1,...,\mu_n-c-1,...$ where $\mu_0,\mu_1,...,\mu_n,...$ are the eigenvalues of (2). If I calculate the eigenvalues of $A_s$ then i can calculate $\lambda$. 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 I am getting somewhere or I am just waisting everyones time...

sorry for bad english and thanks for help

 

[Evalesco]

It sounds like you have a good idea of what you need to do, but could use some help in the technical details. The approach of using the WKB method seems like a good idea, but applying boundary conditions may be tricky. You can try using numerical methods to solve the problem, such as finite difference or finite element methods, which are often used to solve ODEs. You can also try looking up numerical methods for solving quasi-linear ODEs. Additionally, it might help to consult a textbook on differential equations, as they will often provide helpful tips and methods for solving problems like this.

 

[tacman]

First of all, it is important to note that the stability of a solution to a quasi-linear ODE is determined by the eigenvalues of the linearized equation around that solution. In this case, the linearized equation is (2) and the eigenvalues are given by \mu. So, in order to determine the stability of the solution to (1), we need to find the values of \mu for which the linearized equation has non-trivial solutions.

One approach to finding these values is to solve the linearized equation with \mu=0, as you mentioned. This will give us the critical value of \lambda, denoted by \lambda_c, at which stability is lost. This means that for \lambda<\lambda_c, the solution to (1) is stable, and for \lambda>\lambda_c, it is unstable.

To solve for \lambda_c, we can use a numerical method such as the shooting method or the method of continuation. These methods involve solving the linearized equation (2) with \mu=0 for a range of values of \lambda and checking for non-trivial solutions. The value of \lambda at which the first non-trivial solution is found is \lambda_c.

Another approach, as you mentioned, is to use the operator A_c. This method involves finding the eigenvalues of A_c and then using them to determine the values of \lambda for which the linearized equation has non-trivial solutions. This can be a more efficient method, but it requires knowledge of the eigenvalues of A_c, which may not always be easy to find.

As for the WKB method, it can also be used to find the eigenvalues of the linearized equation. However, it can be a bit more involved and may not always give accurate results.

In summary, to determine the stability of the solution to (1), you can use any of the above methods. It is always a good idea to check the results using different methods to make sure they are consistent. I hope this helps clarify your approach and good luck with your calculations!