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Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$.

$$ \begin{cases} \tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\gamma_{uv}\right)\\ \tau_v&=G\left( \gamma,\gamma_u,\gamma_v,\gamma_{uv}\right)\\ \tau^2&=H\left( \gamma,\gamma_u,\gamma_v,\gamma_{uv}\right) \end{cases} $$

being $F,\:G$ and $H$ specific, independent non-linear functions of $\gamma$ and its first derivatives, but linear in $\gamma_{uv}$. $\gamma(u,0)$ and $\gamma(0,v)$ are given. The question is what are the conditions that we need to impose or need to be satisfied for the existence of (non-trivial-)solutions.

In principle we can obtain a third order equation imposing $F_v=G_u$, solving for $\gamma$ and then simply integrating $\tau$ directly. However, such a third order equation will need additional conditions, and how to guarantee that the solutions will also satisfy the remaining relation ?

EDIT: The question comes from a problem in mechanics, with the specific forms of $F,\:G$ and $H$ as follows

\begin{align} F&=\alpha\:\frac{\gamma_{uv}}{\sin\gamma}+\frac{\gamma_u}{\sin\gamma}\left( \alpha_v+\beta\:\frac{\gamma_v}{\sin\gamma}\right) \\ -G&=\beta\:\frac{\gamma_{uv}}{\sin\gamma}+\frac{\gamma_v}{\sin\gamma}\left( \beta_u+\alpha\:\frac{\gamma_u}{\sin\gamma}\right) \\ H&=\frac{\gamma_{uv}}{\sin\gamma}-\alpha\beta\:\frac{\gamma_u\gamma_v}{\sin^2\gamma} \end{align}

being $\alpha=\alpha(u,v)$ and $\beta=\beta(u,v)$ arbitrary smooth functions.

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    $\begingroup$ Depends a lot on what $F,G,H$ are. If you have specific formulas, that might be helpful. $\endgroup$
    – Deane Yang
    Commented Nov 14, 2023 at 19:01
  • $\begingroup$ @DeaneYang Thank you. Please see edit. $\endgroup$
    – Daniel Castro
    Commented Nov 15, 2023 at 12:28
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    $\begingroup$ Some observations (but I'll let you do the calculations): You can eliminate $\tau$ in different ways. I'll call the equations (1),(2),(3). First, take the square root of (3), differentiate it with respect to $u$ and subtract (1). Second, do the same with (3) and (2). Differentiate (1) with respect to $v$, differentiate (2) with respect to $u$, subtract. This yields 3 equations for one function $\gamma$, which is an overdetermined system. Unless your functions $F$, $G$, $H$ satisfy certain conditions, there will be no solutions to this system. Perhaps you could calculate the 3 equations. $\endgroup$
    – Deane Yang
    Commented Nov 15, 2023 at 20:23
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    $\begingroup$ Can you say what you mean by $\alpha$ and $\beta$ being `arbitrary smooth functions'? Are they part of the unknowns or are they specified in advance? Note that in the case $\alpha \equiv\beta\equiv 0$, the function $\tau$ would be a constant $c$, and $\gamma$ would just have to satisfy $\gamma_{uv} = c^2\,\sin\gamma$. However, this is very exceptional. On an open set in $uv$-space where $\alpha\beta$ is nonvanishing, there is only a finite dimensional space of possible nontrivial solutions. $\endgroup$
    – Robert Bryant
    Commented Nov 15, 2023 at 21:19
  • $\begingroup$ @RobertBryant Thank you. $\alpha$ and $\beta$ are know functions, specified in advance. They are part of an experimental input, and as you rightly say, when they vanish or when they are identical constants one can immediately solve for $\gamma$ and $\tau$. $\endgroup$
    – Daniel Castro
    Commented Nov 16, 2023 at 11:29

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