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This is a cross-post.

In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system

\begin{align} \frac{\partial}{\partial y}\left(\frac{1}{\beta}\frac{\partial\alpha}{\partial y}\right)&=K\:\alpha\beta\\ \frac{\partial}{\partial x}\left(\frac{1}{\alpha}\frac{\partial\beta}{\partial x}\right)&=-K\:\alpha\beta, \end{align}

with $K$ a fixed, real, positive constant. We typically set Goursat data along the axes consisting of $\alpha(x,0)$ and $\beta(0,y)$, so the first question is

Does there exist a unique solution? (At least on points sufficiently close to the initial lines)

(moreover, there should be no reason to restrict $K$ to be a constant, but the same answer may apply if it's a smooth function.)

Setting aside the initial conditions, is it possible to write down some analytic solutions with $\alpha$ and $\beta$ other than polynomials?

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  • $\begingroup$ Any chance you’d be interested in setting the initial data along the line $y=x$ or any other line but the axes? $\endgroup$
    – Deane Yang
    Commented Oct 10, 2021 at 22:20
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    $\begingroup$ @DeaneYang Thank you. Non-characteristic lines are not our primary focus, but any suggestions even in that case would be helpful. $\endgroup$
    – Daniel Castro
    Commented Oct 10, 2021 at 22:25
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    $\begingroup$ It was just a hopeful but not really serious question. The noncharacteristic initial value problem is well-posed, and there exists a unique solution in a neighborhood of the line. It also seems possible that if the initial data is sufficiently small, a globl solution exists. $\endgroup$
    – Deane Yang
    Commented Oct 11, 2021 at 1:44
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    $\begingroup$ Are you sure you are giving enough initial data? For a system like this, usually local existence and uniqueness is independent of lower order perturbations. So consider the simplified equation where $K = 0$. Suppose $\alpha(x,0) = a_0$ and $\beta(0,y) = b_0$, then any pair of linear functions of the form $\alpha(x,y) = a_0 + \gamma y$ and $\beta(x,y) = \beta_0 + \delta x$ would solve the system. I expect similar counter examples can be found after restoring $K > 0$. $\endgroup$
    – Willie Wong
    Commented Oct 11, 2021 at 1:51
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    $\begingroup$ Daniel, just for the sake of correctness, It would be advisable to add a remark to point out that you have already asked the same question several days ago and no one has answered. $\endgroup$
    – Daniele Tampieri
    Commented Oct 11, 2021 at 7:56

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