Hi,

The short answer to your question is: there isn't one that can be applied in general. There may be only "satisficing" ones, for specific nonlinear (discrete or continuous), stable or unstable, equilibrium or non-equilibrium, single or multiple structural equilibria containing dynamical equations (problems) you either formulate yourself, or come across having been formulated by others.

There are significant methodological and epistemological questions involved in your question; as well as yet unsolved (possibly unresolvable) mathematical problems.

Good luck!

I advise you to read the following paper , And see which one is closer to your problem. In fact we can not say this or that method is the best

1-=Exact travelling wave solutions for two nonlinear evolution equations using the improved -expansion method

A. Elhanbaly, M.A. Abdou

Journal:

Mathematical and Computer Modelling

2-A series of new exact solutions of nonlinear evolution equations

M. A. Abdou

Journal:

Numerical Methods for Partial Differential Equations

3-Existence of mild solutions for semilinear evolution equations with non-local initial conditions

Qing Liu, Rong Yuan

Journal:

Nonlinear Analysis: Theory, Methods & Applications

Dear Dimitrios S. Dendrinos,

Thank you very much for your outstanding answer and good thinking. Your suggestion will make me find a new dimension for my future research work.

As others have commented, there is no 'magic bullet' with regard to solving nonlinear evolutionary equations. I mention a few problem types and possible solutions below.

For most problems with solutions that do not exhibit shocks or steep gradients, or become close to being singular, standard Method Of Lines (MOL) with a variable step-size Backwards Differentiation (BD) integrator will usually suffice.

Where solutions exhibit 'shocks' or 'steep gradients', a 5th order Weighted Essentially Non-Oscillatory (WENO) method will often be a good choice.

Where the problem includes an irregular spatial domain, use of Radial Basis Function (RBF) or Finite Element (FE) methods may be necessary.

Where the problem becomes singular at some point in the solution, well established LSODES is a good choice for the numerical integrator.

However, there are many physical problems, not mentioned above, or that contain more than one of the above difficulties, and these may require special purpose solvers. For example, solutions of celestial problems where very long-term solutions are required need special integration methods to keep errors to a minimum.

I have included a number of solution examples for a range of PDE problems on my website www.pdecomp.net. The computer code is free to download.

With kind regards,

Graham W Griffiths

Dear Graham W Griffiths,

Thanks for your informative suggestions.

This a very broad area. For an evolution equation not involving spatial partial derivatives - the theory of ordinary differential equations in finite dimensional space is very rich in this are. However, because of nonlinearities they techniques required, e.g., topological methods, fixed point theorems, etc., vary from equation to equation depending on the properties of the non-linear terms. But in general if the vector field defending the equation is smooth enough and bounded the solution exist globally and much can be said.

However, in the case involving spatial partial derivatives - the underlying space for such an equation is a function space Sobolev spaces of distributions and the approach it to establish a weak solution and they establish uniqueness and regularity for such weak solutions. Neither is by no means guaranteed and is highly dependent on the details of the nonlinearity. The other issue with these equations, which model much of physics, is "blowing up" having no global solutions. However, most of the equations of mathematical physics are of this form, from the Kortweg-deVries equation which a lot is known to the Navier-Stokes Equation which a lot is still unknown above dimension two.

https://www.claymath.org/sites/default/files/navierstokes.pdf

BTW if you solve this - you be a million dollars richer since it is one of the Clay Mathematics institute's Millennium problems.

https://www.claymath.org/millennium-problems

Even when there is no linearity, i.e., f' = Af where A is a linear partial differential operation. The conditions of A are very important in understanding the behavior of such a solution. This equation was the impetus for the development of the theory of semi-groups of operators by Hille, Stone, Yosida, Kato and others.

Here is a good introductory account.

https://www.math.ucdavis.edu/~hunter/notes/nonlinev.pdf

For the theory of semi-groups and related functional analytic concpets

https://www.zuj.edu.jo/download/functional-analysis-k-yoshida-pdf/

And for a very good reference on non-linear PDE including evolution equations of course Michael Taylor's wonderful book.

https://www.amazon.com/Partial-Differential-Equations-III-Mathematical/dp/1441970487/ref=sr_1_1?Adv-Srch-Books-Submit.x=40&Adv-Srch-Books-Submit.y=7&dchild=1&keywords=Partial+differential&qid=1631803454&refinements=p_27%3ATaylor&s=books&sr=1-1&unfiltered=1

Dear Truman Prevatt,

Thanks a lot for your informative suggestions.

check these might be helpful

https://www.hindawi.com/journals/amp/si/627647/