If NDsolve cannot solve it you can try changing methods (option to NDsolve). "monte carlo" I think has a good chance of solving where other methods fail. Mathematica also comes with "equation trekker" you might find that useful.
Your pde is only using differential of y? You could see what happens if you give x a constant value.
You have at least one typo ... f(0,t) should be f[0,t]. I suspect also you have a second typo because your original equation indicates D[ f[x,t], x] on the left of the equality. You have (1/4f) and if you would clarify if you intended (1/4)f or (1/(4f)) since that "changes everything" (for solving by hand). (It would help if you mentioned what book or subject, title of chapter, so anyone can can help further without solving "a difficult problem" and later finding it was the wrong question please).
(Nonlinear ... there are system of PDE method, perturbation method, numerical, and (well, an endless list of table of solutions to every kind of nonlinear pde in CRC books), methods that require conditions and inside knowledge of the physics going on).
"Practically all nonlinear PDEs must be solved by numerical methods, and, in fact, most realistic models in physics, chemistry, biology, and so forth, are nonlinear in nature.
Excerpt From: Stanley J. Farlow. Partial Differential Equations for Scientists and Engineers. Apple Books. https://books.apple.com/us/book/partial-differential-equations-for-scientists-engineers/id490945639"
