It is often possible to write down some special solutions explicitly in terms of elementary functions (though it is rarely possible to describe all solutions like this). One way of finding such explicit solutions is to reduce the equations to equations of lower dimension, preferably ordinary differential equations, which can often be solved exactly. This can sometimes be done using separation of variables, or by looking for highly symmetric solutions.
Some equations have several different exact solutions.
Read more about this topic: List Of Nonlinear Partial Differential Equations, Methods For Studying Nonlinear Partial Differential Equations
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