you can use some analytical methods like: perturbation method, ADM, DTM,...etc.

see attached file

One of the main important method for this case is inverse scattering method.

The most applicable method is to make some symmetry ansatz (like a similarity form), and transform the PDE to an ODE, which hopefully is exactly solvable. I don't think the inverse scattering method has much practical value; a direct numerical solution will most likely be much better in most cases (unless you are investigating very special effects).

you can use several analytical methods such as Duan-Adomian method, VIM, HPM, reduce order methods,......

good luck

The inverse scattering method gave analytical solutions to the Korteweg de Vries and the Sine-Godon equations, and more.  A numerical solution won't identify soliton solutions.  That's why Kruskal and al.  searched for analytical methods, in order to understand what they saw numerically.

Claude ~

That is what I referred to as very special effects, caused by the uncommonly large number of conservation laws in some very special systems.

Single-soliton solutions can be found more easily as (the limiting case of) a nonlinear plane wave, which is an example of the symmetry ansatz solutions I mentioned. There is a discussion of this on another Q&A thread just now, cf. the link below.

https://www.researchgate.net/post/Does_anyone_have_information_of_Tanh_or_Sech_methods_for_finding_exact_solutions_of_some_nonlinear_partial_differential_equations

The ISM is a very general method that generates all the solutions, along with their nature.  It shows that there is only two of them.  In a numerical calculation approach, that entails trying every possible initial conditions, with impredictable result owing to the very nonlinearity.  The drawback is that there is no systematic way to get the associated linear system, but if the equation belongs to the known ones, the solution can be found in the literature.  It works mainly in only 1+1 dimensions too.

Firstly, you should be more specific about the shape and kind of the non-linear equation you want to solve. 

Second step is to simplify equation by using approximations based on justified assumptions.

Next, every case should be solved analytically.

In the end, comparison should be made with asymptotics or/and numerical solution.

Note: Errors should be estimated seperately.

Goodluck.

There is no special method for solving a nonlinear partial differential equation , because in general we do not know how. But very little of equations the variable separation method is used to obtain  an ordinary differential equation.

 For strong nonlinear differential equations ,you could use HAM,ADM,LASPM

Hi Mr.Morteza,

I suggest you this book, Hope that can help you

http://en.bookfi.org/book/1452114

Good Luck.

Ali M. Hourina: I am very grateful for this book. I have for long be searching for a comprenhesible book on nonlinear partialdifferential equations. Thank you one again.

Some nonlinear PDE's can be solved using the results on inverse spectral problems for ordinary differential equations. E.g. I have written an article for the Korteweg–De Vriez equation. I suggested a method of finding solutions tending to some constants at infinities...

There is no exact solution to nonlinear pde.

Well, for non-linear problems there is no standard procedure which could be followed. However, some similarity transforms exist for specific problems as quoted in the book of PDE by Evans. One can also convert specific problems to linear ones (e.g. Cole-Hopf transformations convert Burgers Eqn into a linear pde). Again all the transforms and methods are problem specific mostly. In cases where all these  analytic methods fail, approximate methods provide an alternative means to achieve approximate solutions. In approximate solution ADM method give a series of polynomials as approximations. FEM also gives good results. 

These equations can in many cases be solved via  suitable transformations leading to ODEs. I admit some ingenuity may be involved.