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Solutions of a Gauss–Codazzi-like system of nonlinear PDEs

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_{ …
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