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Given a PDE

$ f e^2 \frac{\partial f}{\partial x}- e f^2 \frac{\partial f}{\partial y} + M_1 f^4 + M_2 f^2 + M_3=0 $

Note that $M_1$ , $M_2$ and $M_3$ are functions of $\cos (x-y)$ and $\sin (x-y)$

e is a constant

How could one solve this PDE ? Any suggestion

Shuchang
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ahmed1
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    a method used for non-linear PDEs is a generalisation of fourier transform known as inverse scattering method – Nikos M. Aug 15 '14 at 14:46
  • another one is approximate solution about a solution of the linearised PDE (in a sense of approximation or perturbation) – Nikos M. Aug 15 '14 at 14:48
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    A general method to solve this kind of PDE is the so-called Lagrange-Charpit method which follows from the characteristics method. (See here http://math.stackexchange.com/questions/897299/initial-value-problem-for-non-linear-partial-differential-equation-y-x2-k-y-t/897605#897605) If you write down all the equalities, I'm afraid you come up with something completely unpleasant. – Dmoreno Aug 15 '14 at 17:36
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    inspired by @Dmoreno ' comment, the method of characteristics, seems to be better suited for quasi-linear PDEs – Nikos M. Aug 15 '14 at 20:33
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    Most certainly, the method of characteristics is what you want. – Igor Khavkine Aug 16 '14 at 13:48

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