I was wondering on the solution of the equation $$\nabla^2P(\vec r)=v(\vec r)P(\vec r)^2\phantom{.......}(1)$$ Or more simply, if there exists a coordinate system where: $$\nabla^2P(\vec r)=P^2(\vec r)$$ Especially considering that if $\vec r=\xi_k$, a general orthonormal set of coordinates, then the Laplacian of such a coordiante system in Curvilinear coordinates is: $$\nabla^2f(\vec r)=\sum_i\frac{1}{\prod_k\sqrt{\hat{\bar g}_{kk}}}\frac{\partial}{\partial \xi_i}\left(\frac{\prod_k\sqrt{\hat{\bar g}_{kk}}}{\hat{\bar g}_{ii}}\frac{\partial f}{\partial \xi_i}\right)$$
2026-03-31 05:02:31.1774933351
Nonlinear-Variation of Helmholtz Equation
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