A cute little system of nonlinear PDEs

Question

I am wondering about the solutions to the following system of PDEs. Suppose we have functions a(x,y,z), b(x,y,z), and c(x,y,z) and the following equations: $$\partial_x[aa^*+bb^*+cc^*]=2(a^*\partial_xa+b^*\partial_xb+c^*\partial_xc),\\ \partial_y[aa^*+bb^*+cc^*]=2(a^*\partial_ya+b^*\partial_yb+c^*\partial_yc),\\ \partial_z[aa^*+bb^*+cc^*]=2(a^*\partial_za+b^*\partial_zb+c^*\partial_zc).$$ Now, clearly if each of $a$, $b$, and $c$ are purely real and differentiable functions then these equations are satisfied. Are there any other solutions, that is, for complex a b and c?

Another way to write it would be: $\vec{\nabla}(|a|^2+|b|^2+|c|^2)=2(a^*\vec{\nabla}a+b^*\vec{\nabla}b+c^*\vec{\nabla}c)$

Answer 1

Your question seems quite complex, so I will try to illustrate the main ideas by a simplified version of your question, that is : "What would be the complex solutions to the equation $${\partial \over {\partial {x_i}}}\left( {{f^*}({x_1},{x_2},{x_3})f({x_1},{x_2},{x_3})} \right) = 2{f^*}({x_1},{x_2},{x_3}){\partial \over {\partial {x_i}}}\left( {f({x_1},{x_2},{x_3})} \right),i \in \left\{ {1,2,3} \right\}$$". To answer this, let's decompose the function into real and imaginary parts as $$f({x_1},{x_2},{x_3}) = {f_{\rm{R}}}({x_1},{x_2},{x_3}) + i{f_{\rm{I}}}({x_1},{x_2},{x_3})$$Replacing this equation in the main equation and separating the real and imaginary parts leads to $$\left\{ \matrix{ {\partial \over {\partial {x_i}}}\left( {{f_{\rm{R}}}^2({x_1},{x_2},{x_3}) + {f_{\rm{I}}}^2({x_1},{x_2},{x_3})} \right) = 2\left( \matrix{ {f_{\rm{R}}}({x_1},{x_2},{x_3}){\partial \over {\partial {x_i}}}\left( {{f_{\rm{R}}}({x_1},{x_2},{x_3})} \right) + \cr {f_{\rm{I}}}({x_1},{x_2},{x_3}){\partial \over {\partial {x_i}}}\left( {{f_{\rm{I}}}({x_1},{x_2},{x_3})} \right) \cr} \right) \cr 0 = {f_{\rm{R}}}({x_1},{x_2},{x_3}){\partial \over {\partial {x_i}}}\left( {{f_{\rm{I}}}({x_1},{x_2},{x_3})} \right) - {f_{\rm{I}}}({x_1},{x_2},{x_3}){\partial \over {\partial {x_i}}}\left( {{f_{\rm{R}}}({x_1},{x_2},{x_3})} \right) \cr} \right.$$Note that the real part is valid for any differentiable pair of real functions $\left\{ {{f_{\rm{R}}}({x_1},{x_2},{x_3}),{f_{\rm{I}}}({x_1},{x_2},{x_3})} \right\}$The complex part enforces $${f_{\rm{R}}}({x_1},{x_2},{x_3}){\partial \over {\partial {x_i}}}\left( {{f_{\rm{I}}}({x_1},{x_2},{x_3})} \right) = {f_{\rm{I}}}({x_1},{x_2},{x_3}){\partial \over {\partial {x_i}}}\left( {{f_{\rm{R}}}({x_1},{x_2},{x_3})} \right)$$ which has plenty of solutions (for example ${f_{\rm{I}}}({x_1},{x_2},{x_3}) = \alpha {f_{\rm{R}}}({x_1},{x_2},{x_3})$)

Answer 2

Yes. Simply let $a = b^*$ and $c$ be real.

Note that your equation is equivalent to stating that $$ \Im (a^* \nabla a + b^* \nabla b + c^* \nabla c) = 0 $$ Writing $a_R$ for the real part of $a$ and $a_I$ for the imaginary part, this is $$ a_R \nabla a_I - a_I \nabla a_R + b_R \nabla b_I + \cdots = 0 $$ which is a collection of three PDEs for 6 unknown real valued functions.

This has plenty of solutions: for example another set of solutions can be had if $a_I = \alpha a_R$ for another real number $\alpha$, and similarly for $b,c$.