Could you please help me to solve the following equation?
\begin{equation} u_{yyyy}+u_{xy}-a\,\left(u\,u_y\right)_y\,=\,0 \end{equation}
Where \begin{equation} u\,=\phi^{\alpha} \sum_{k\,=\,0}^{\infty}u_k\,\phi^{k},\\ \\ \text{where}\ \phi=\phi(x,y)\ \ \text{and}\ \ u_j=u_j(x,y) \end{equation}
What is $u(x,y)$? I was trying, but it did not work.
I have found two asymptotic solutions but not the general solution, and I'm not sure how these solutions help you with the summation notation format of the solutions, so I can only hope this helps a little bit. I found them using a group theory algorithm, expanding the last term to make life easier.
$$\frac{\partial ^4 u}{\partial y^4}+\frac{\partial}{\partial y}\frac{\partial u}{\partial x}-\frac{a}{1}\bigg(\frac{\partial u}{\partial y}\bigg)^2-\frac{au}{1}\frac{\partial^2u}{\partial y^2}=0$$
This is a group theory application with roots in the Orbit-Stabilizer Theorem, a counting theorem. Simply count the variables in each monomial. If it's in the numerator it's positive, if in the denominator it's negative. Set the results of each monomial equal to each other and find independent algebraic solutions of u.
$$u=4y=u-x-y=a+2u-2y$$ $$u_1=-2y-a$$ $$u_2=y-a-x$$ Reverse the previous process, letting the coefficients be the exponents, and include a constant. $$u_1(x,y)=\frac{C}{ay^2}$$ $$u_2(x,y)=\frac{Dy}{ax}$$
Taking selected derivatives and plugging them back into the equation gives the values of C=12 and D=-1. Therefore your special solutions are $$u_1(x,y)=\frac{12}{ay^2}$$ $$u_2(x,y)=\frac{-y}{ax}$$ These are easily verified. Again, I don't know how to reconcile these with your required format so this may be of limited use to you.