Let $u$ be a solution to the following pde : $$\partial ^2_{tt} u = \partial_{xx} (u^{3/2}),$$ with $u(0,x)=\phi (x)\; ; \: \partial _t u(0,x)= \psi (x).$
In a paper see here M. Dreher says on page 5 (25) that we can hope that such a solution can be written in the form : $$u(t,x)= \phi (x) g(t,x) + \psi (x) h(t,x)$$ with $g(0,x)=1, \; h(0,x)=0, \; \partial _t g(0,x)=0, \; \partial_t h(0,x)=1$ which is not obvious for me. I didn't manage to find it natural, or at least motivated.
Can someone enlighten me with explaining why we hope for this ansatz to be a good one for the solutions $u$ ?
In the paper the author refers to an article of P. D'ancona and R. Manfrin (ref [4]) telling that the idea comes from here, but I didn't manage to find the article on the internet, nor in my university library. If someone manage to find the article I would like to see it if possible.
Thank you,
PS. Note that in the article the setting is a bit more general but that does not change my question.