Solvability and adjoint equation?

[jollage]

Hi all,

I don't understand the following procedure discussing the solvability of the differential equation. Please have a look and comment.

The operator $\mathcal{L}$ is defined as $\mathcal{L}=\mathcal{L}(\phi_0;k,\omega,X)$, where $\phi_0$ is the variable and $k,\omega,X$ are the parameters.

Now take the derivative w.r.t. k of the homogeneous equation $\mathcal{L}=0$, we have $\mathcal{L}(\partial_k\phi_0;k,\omega,X)=-\mathcal{L}_k(\phi_0;k,\omega,X)-\mathcal{L}_{\omega}(\phi_0;k,\omega,X)\partial_k \omega$, where it assumes $\omega$ is dependent on $k$ while $X$ independent on $k$.

Then the author writes that the solvability of the above differential equation requires that the right hand side be orthogonal to the homogeneous solution to the adjoint equation of $\mathcal{L}=0$.

I've came across such solvability criterion now and then (I guess it is common knowledge), but never truly understand what it really means and where it comes from. Also, what's the role of the adjoint equation? Could you help me? Thanks a lot in advance.

 

[jvicens]

Hello,

Thank you for bringing this question to the forum. The procedure you are describing is a common approach in solving differential equations. Let me explain it in more detail.

First, the operator \mathcal{L} is defined as a function of the variable \phi_0 and the parameters k, \omega, and X. This means that for a given set of values for k, \omega, and X, the operator will produce a unique result for any given value of \phi_0.

Next, the author takes the derivative of the homogeneous equation \mathcal{L}=0 with respect to k. This results in a new equation, where the left hand side is the derivative of \mathcal{L} with respect to k, and the right hand side is a combination of the derivatives of \mathcal{L} with respect to k and \omega.

Now, the author states that in order for this differential equation to be solvable, the right hand side must be orthogonal to the homogeneous solution of the adjoint equation of \mathcal{L}=0. The adjoint equation is essentially the transpose of the original equation, and its homogeneous solution is the solution to the original equation. By requiring the right hand side to be orthogonal to this solution, we are essentially ensuring that the solution to the original equation is not affected by changes to the parameters k and \omega.

In other words, the solvability criterion is a way to ensure that the solution to the differential equation remains unchanged when the parameters are varied. This is important because it allows us to find a solution that is independent of the specific values of the parameters, making it a more general solution.

I hope this helps to clarify the procedure and the role of the adjoint equation in determining the solvability of the differential equation. If you have any further questions, please let me know. Thank you.