Exact solutions to some nonlinear PDEs, travelling profiles method

We suggest finding exact solutions of equation: @u @t = ( @ m @x m u) p , t � 0, x 2 R, m, p 2 N, p > 1, by a new method that we call the travelling profiles method. This method allows us to find several forms of exact solutions including the classical forms such as travelling-wave and self-similar solutions.


Introduction
Consider the following equation : where A x u is a nonlinear differential operator.
For seeking exact solutions to nonlinear PDEs (1.1), there are three approaches in general: 1-Travelling-wave solutions (see for example [4,10,17]): The principle of this method is to seek a solution in the form : where the function u is solution of following differential equation : 2-Self-similar solutions (see for examples [4,8,10,14,16]): This method is largely used, its principle is to seek a solution in the form : EJQTDE, 2008 No. 15, p. 1 where β and γ are some constants, the function u is determined by the differential equation : There exists also a general form of self similar solutions in the form ), (1.3) where ϕ(t) and ψ(t) are chosen for reason of convenience in the specific problem.
In this paper we propose a new approach to find exact solutions to some nonlinear PDEs in the form: This equation engenders many vell known problems such as the porous medium equation (PME) for m = 2 (see [1,8,9]).
The new approach which we will present is called the travelling profiles method (TPM).
2 The travelling profiles method (TPM): The principle of this method is to seek the solution of the problem (1.4) under the form where ψ is in L 2 , that one will call the based-profile.The parameters a (t) , b (t) , c (t) are real valued functions of t.
therefore, we obtain three orthogonality equations which are read as: where ., . is the inner product in L 2 space.The PDE (1.4) is then transformed into a set of three coupled ODE's : (2.4)

A priori estimates of solutions :
Let: the subspace of L 2 generated by associated functions to ψ at the moment t.
From relations (2.3), it is deduced that ∂u ∂t − ( ∂ m ∂x m u) p is orthogonal to subspace V t .In particular we have ∂u ∂t V t , then ∂u ∂t − ( ∂ m ∂x m u) p , ∂u ∂t = 0, thus if also ( ∂ m ∂x m u) p belongs to V t then the method provides us a weakly exact solution, which is written under the form u (x, t) = c (t) ψ x − b (t) a (t) . (2.5)

Exact solutions:
Theorem : is an exact solution of problem (1.4), if the based profile ψ is a solution of following differential equation where α, β, γ ∈ R, with α, β, γ = 0, in this case, the coefficients c (t) , a (t) , b (t) are given by : is an exact solution to equation (1.4).In this case the term ( d m dξ m ψ) p can be expressed as a linear combination of functions ψ, ξψ ξ , and ψ ξ .In other words we have ( d m dξ m ψ) p = αψ + βξψ ξ + γψ ξ , for α, β, γ ∈ R. The coefficients c (t) , a (t) , b (t) are obtained as follow: When one replaces ( d m dξ m ψ) p by the combination αψ + βξψ ξ + γψ ξ in system (2.4), we obtain: with where ., . is the inner product in L 2 .
The matrix M in the system (2.7) is symmetric and invertible, therefore (2.7) can be written under the form: From (2.8) we have , with K 0 , K 0 constants. (2.9) If we replace (2.9) in (2.8), then we deduct (2.6).

Example:
Let us consider the equation If we seek an exact solution like u (x, t) = c(t)ψ x−b(t) a(t) , the based-profile ψ must verify the following ODE: (ψ ξ ) Then we obtain an exact solution to equation (2.10) under the form: where c(t), a(t), and b(t) are given (from 2.6) by: 3 Some particular forms: In our approach we can find the particular forms of well-known solutions such as travelling-wave and self-similar solutions.

Travelling-wave solutions :
If we seek a solution to equation (1.4), like we obtain a class of travelling-wave solutions, where the based profile ψ is solution of following ODE: