A generalized exponential transform method for solving non-linear evolution equations of physical relevance
Section snippets
Introduction – general comments
Mathematical modelling of physical systems often leads to non-linear evolution equations in the general form ut = K[u], where K[u] is a locally defined function of u and its x-derivatives. The best-known example (also from historically point of view) is the Korteweg de Vries equation [1] and its varieties (cylindrical KdV, modified KdV, generalized KdV) modeling the propagation of weakly non-linear waves in dispersive media. Further, the equation represents the starting point to investigate
Description of the method
Consider a given nPDE in its two variables x and t which describes the dynamical evolution of a wave form u(x, t), u : Ω → in a domain Ω ⊆ d, d ⩾ 2 and t, so that the mapping holds for u : Ω × + → , + = {t ∈ : t > 0}, Ω ⊆ d:Firstly, in our search for stationary solutions of (2.1), we introduce a travelling frame of reference ξ = x − λt (whereby λ is a constant to be determined in a later step; in general wave theory or solitary propagation λ means the velocity) to transform (2.1) into an nODE, say
Examples, calculations and results
Example 1 A non-linear evolution equation of fourth order The scaled equation [34], [38] in (1 + 1) dimension under consideration is given by:whereby u(x, t) is a field describing a wave propagation depending on time t so that the mapping holds for u : Ω × + → , + = {t ∈ : t > 0}, Ω ⊆ d. u(x, t) should be continuous and differentiable at least four times.
We are looking for class of solutions for which u = F(x, t), F ∈ C4(D), D ⊂ 2 holds. Performing the steps considered in Section 2 with n = 2, the following NHAS has to be
Properties of some selected class of functions
We start the analysis with the solution function (3.4b) where a graphical sketch can be seen by Fig. 1. To be a regular function, one has to claim the restriction . For the limit case t → 0 as an initial condition, one can see the behaviour by limx→±0 Re[uA,I(x, 0)] remaining a finite expression, otherwise in the case of limx→±∞Re[uA,I(x, 0)], the function is indefinite. In a considered domain the function is not stable, provided byTherefore, one
Conclusion remarks
In this paper, new class of solutions of some (1 + 1) non-linear evolution equations of higher order by using a new direct algebraic approach, the so called generalized exponential transform method (GETM) could derived.
It is well known that class of solutions of non-linear evolution equations can be expressed as finite series in terms of special functions, e.g. tanh-functions, Weierstrass and Jacobian functions.
All these different algebraic approaches are useful if one assume the possibility to
References (40)
- et al.
Wave Motion
(1985) Chaos Solitons Fract.
(2006)Appl. Math. Comput.
(2008)- et al.
Chaos Solitons Fract.
(2003) - et al.
Chaos Solitons Fract.
(2004) - et al.
Chaos Solitons Fract.
(2004) Phys. Lett. A
(1996)- et al.
Phys. Lett. A
(1998) - et al.
Phys. Lett. A
(1998) Phys. Lett. A
(2001)
Chaos Solitons Fract.
Phys. Lett. A
Chaos Solitons Fract.
Chaos Solitons Fract.
Chaos Solitons Fract.
Chaos Solitons Fract.
Physica D
Chaos Solitons Fract.
Appl. Math. Comput.
Physica D
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