Abstract
The Lax equation is introduced on a time-space scale. The viscous Burgers and the nonlinear Schrodinger dynamic equations on a time-space scale are deduced from the Lax equation by using the Ablowitz-Kaup-Newel-Segur-Ladik method. It is shown that the Burgers equation turns to the heat equation on a time-space scale by the Cole-Hopf transformation. Further, using the separation of variables, we deduce the formula for solutions of the boundary value problem for the heat and Burgers equation on a time-space scale in terms of Fourier series by Hilger exponential functions.
Highlights
In Lax [ ] introduced the linear operator equation equivalent to the nonlinear Korteweg-de Vries (KdV) equation that describes the traveling solitary waves
The importance of Lax’s observation is that any equation that can be cast into such a framework may have remarkable properties of the KdV equation, including the integrability and an infinite number of local conservation laws
There are several other methods to generate the integrable hierarchy of nonlinear dynamic equations: Ablowitz-Kaup-Newel-Segur (AKNS) method [ ], Gelfand-Dickey method [ ], Ablowitz-Kaup-Newel-Segur-Ladik (AKNSL) method [ ], which is the extension of AKNS method on difference equations
Summary
In Lax [ ] introduced the linear operator equation equivalent to the nonlinear Korteweg-de Vries (KdV) equation that describes the traveling solitary waves. If θ (t, x) is t-regressive, the nabla t-exponential function eθ (t, t , x) on a time scale T can be defined as the unique solution of the initial value problem (see [ , ]). If θ (t, x) is x-regressive, the nabla x-exponential function eθ (t, x, x ) on a space scale X is defined as the unique solution of the initial value problem e(θx)(t, x, x ) = θ (t, x)eθ (t, x, x ), eθ (t, x , x ) =. ). Consider the initial value problem for the heat equation on a time-space scale v(t)(t, x) = p v(xx)(t, x), v(t , x) = φ(x), x ∈ X, t, t ∈ T. One can get the well-known formula for the solution of initial value problem for the classical viscous Burgers equation. R(t, y)∇y , ν(y) where the variation of a parameter formula (see [ ]) is used for the solution of the first order dynamic equation on a space scale
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