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I want to solve the PDE $$\frac{\partial u}{\partial t}+x_1(x_2-x_3) \frac{\partial u}{\partial x_1}+x_2(x_3-x_1) \frac{\partial u}{\partial x_2}+x_3(x_1-x_2) \frac{\partial u}{\partial x_3}=\sum_{i=1}^3 \alpha_i \frac{\partial f}{\partial x_i}, \tag{1} $$ where $\alpha_1,\alpha_2,\alpha_3$ are constants and $f$ is the function

$$f(\mathbf{x},t)= \frac{\alpha _1 \left(\wp '\left(t;g_2,g_3\right)+x_2x_3 \left(x_2-x_3\right)\right)}{2 \left(\wp \left(t;g_2,g_3\right)-\frac{1}{12} \left(x_1+x_2+x_3\right){}^2+x_2 x_3\right)}+\frac{\alpha_2 \left(\wp '\left(t;g_2,g_3\right)+x_1 x_3 \left(x_3-x_1\right)\right)}{2 \left(\wp \left(t;g_2,g_3\right)-\frac{1}{12} \left(x_1+x_2+x_3\right){}^2+x_1 x_3\right)}+\frac{\alpha _3 \left(\wp '\left(t;g_2,g_3\right)+x_1x_2 \left(x_1-x_2\right) \right)}{2 \left(\wp \left(t;g_2,g_3\right)-\frac{1}{12} \left(x_1+x_2+x_3\right){}^2+x_1 x_2\right)}+\left(\alpha _1+\alpha _2+\alpha _3\right) \left(\zeta \left(t;g_2,g_3\right)+\frac{1}{12} t \left(x_1+x_2+x_3\right){}^2\right). $$

Here $\wp$ and $\zeta$ are the Weierstraß p- and zeta- functions respectively, with the elliptic invariants $$ \begin{align} g_2 &= \frac{(x_1+x_2+x_3)^4}{12}-2 x_1 x_2 x_3 (x_1+x_2+x_3), \\ g_3 &= -(x_1 x_2 x_3)^2+\frac{x_1 x_2 x_3 (x_1+x_2+x_3)^3}{6}-\frac{(x_1+x_2+x_3)^6}{216}. \end{align} $$

From this point onward the invariants $g_2,g_3$ will not be shown explicitly.


My attempt: First, I managed to solve the associated homogeneous PDE $$ \frac{\partial u_h}{\partial t}+x_1(x_2-x_3) \frac{\partial u_h}{\partial x_1}+x_2(x_3-x_1) \frac{\partial u_h}{\partial x_2}+x_3(x_1-x_2) \frac{\partial u_h}{\partial x_3}=0, $$ via the method of characteristics. The solution is given by $$ u_h =\Phi \left( X_1(\mathbf{x},t), X_2(\mathbf{x},t) ,X_3(\mathbf{x},t) \right) $$ where $\Phi$ is an arbitrary function, and $$X_1=\frac{12 x_1 x_2 x_3}{\left(x_1+x_2+x_3\right)^2-12 \left(\frac{\left(\wp'(t)-x_{2}x_{3} \left(x_{3}-x_{2}\right) \right)^2}{4 \left(\wp (t)+ x_{2}x_{3} -\frac{1}{12} \left(x_1+x_2+x_3\right)^2\right)^2}-\wp (t)+ x_{2} x_{3}-\frac{1}{12} \left(x_1+x_2+x_3\right)^2\right)}, \\ X_2 = \frac{12 x_1 x_2 x_3}{\left(x_1+x_2+x_3\right)^2-12 \left(\frac{\left(\wp'(t)-x_{1} x_{3} \left(x_{1}-x_{3}\right) \right)^2}{4 \left(\wp (t)+x_{1} x_{3}-\frac{1}{12} \left(x_1+x_2+x_3\right)^2\right)^2}-\wp (t)+x_{1} x_{3}-\frac{1}{12} \left(x_1+x_2+x_3\right)^2\right)}, \\ X_3 = \frac{12 x_1 x_2 x_3}{\left(x_1+x_2+x_3\right)^2-12 \left(\frac{\left(\wp'(t)-x_{1} x_{2} \left(x_{2}-x_{1}\right) \right)^2}{4 \left(\wp (t)+ x_{1} x_{2}-\frac{1}{12} \left(x_1+x_2+x_3\right)^2\right)^2}-\wp (t)+ x_{1} x_{2}-\frac{1}{12} \left(x_1+x_2+x_3\right)^2\right)}. $$

The final ingredient is a particular solution. Denoting the RHS in Equation (1) above by $R(\mathbf{x},t)$, Duhamel's principle (or the method of characteristics again) suggests that a particular solution is given by $$u_p=\int_0^t R \left( \mathbf{X}(\mathbf{x},t-u),u \right) \mathrm{d} u .$$ I tried computing this with Mathematica and it didn't go well. This is probably because Mathematica seems to be unaware of the elliptic identity $\wp'^2=4 \wp^3-g_2 \wp -g_3$.

I would appreciate help with the evaluation of the integral above, or any other method of obtaining a particular solution of Equation (1).

Thank you!

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  • $\begingroup$ You already know that a solution exists. This is not enough? $\endgroup$ – timur Dec 26 '18 at 0:38
  • $\begingroup$ @timur No, I want to see if I can get it explicitly. $\endgroup$ – user1337 Dec 26 '18 at 9:07

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