q-Analogue of Shock Soliton Solution

It is well known that the Burgers' equation in one dimension could be linearized by the Cole-Hopf transformation in terms of the linear heat equation. It allows one to solve the initial value problem for the Burgers equation and to get exact solutions in the form of shock solitons and describe their scattering. In the present paper we study the q-differential Burgers type equation with quadratic dispersion and the cubic nonlinearity, and find its linearization in terms of the q-heat equation. In terms of the Jackson's q-exponential function we introduce the q-Hermite and q-Kampe-de Feriet polynomials, representing moving poles solution for the q-Burgers equation. Then we derive the operator solution of the initial value problem for the q-Burgers equation in terms of the IVP for the q-heat equation. We find solutions of our q-Burgers type equation in the form of singular and regular q-shock solitons. It turns out that static q-shock soliton solution shows remarkable self-similarity property in space coordinate x.

The q-number corresponding to the ordinary number n is defined as, [1],

where q is a parameter, so that n is the limit of [n] q as q → 1. A few examples of q-numbers are given here:

In terms of these q-numbers, the Jackson q-exponential function is defined as

For q > 1 it is entire function of x and when q → 1 it reduces to the standard exponential function e x . The q-exponential function can also be expressed in terms of infinite product

when q < 1 and

when q > 1. Thus, the q-exponential function for q < 1 has infinite set of poles at

, n = 0, 1, ..

and for q > 1 the infinite set of zeros at

, n = 0, 1, ..

The q-derivative is defined as

and when q → 1 it reduces to the standard derivative D x q f (x) → f ′ (x). Using the definition of the q-derivative one can easily see that

x q e q (ax) = ae q (ax).

(9)

3 q-Hermite Polynomials

We define the q-Hermite polynomials according to the generating function

From this generating function we have the special values

where [n] q ! = [1] q [2] q …[n] q , and the parity relation

By q-differentiating the generating function (10) according to x and t we have the recurrence relations correspondingly

Using operator

so that

relation (15) can be rewritten as

Substituting ( 14) to (18) we get

By the recursion, starting from n = 0 and H 0 (x) = 1 we have next representation for the q-Hermite polynomials

We notice that the generating function and the form of our q-Hermite polynomials are different from the known ones in the literature, [2], [4], [3], [5].

In the above expression the operator

is expressible in terms of the q-spherical means as

Using definition, [1],

. which now we apply for operators, we should distinguish the direction of multiplication. We consider two cases

and

Then, we can rewrite (20) shortly as

First few polynomials are

When q → 1 these polynomials reduce to the standard Hermite polynomials.

Applying D x to both sides of (19) and using recurrence formula (14) we get q-differential equation for q-Hermite polynomials

Proposition 1 We have next identity

Proof 1 By q-differentiating the q-exponential function in x

and combining then to the sum

we have relation

Proposition 2 The next identity is valid

Proof 2 The right hand side of (25) is the generating function for q-Hermite polynomials. Hence expanding both sides in t we get the result.

Proposition 3

Proof 3 we use (30) and relation (19) .

Corrollary 1 If function f (x) is analytic and expandable to power series f (x) = ∞ n=0 a n x n then we have next q-Hermite series

4 q-Kampe-de Feriet Polynomials

We define q-Kampe-de Feriet polynomials as

and from (19) we have the next recursion formula

By recursion it gives

or by notation (24)

First few polynomials are

We consider the q-heat equation

Solution of this equation expanded in terms of parameter k φ(x, t) = e q (νk 2 t)e q (kx) =

gives the set of q-Kampe-de Feriet polynomial solutions for the equation. Then we find time evolution of zeroes for these solutions in terms of zeroes z k (n, q) of q-Hermite polynomials, H n (z k (n, q), q) = 0, (37) so that

For n=2 we have two zeros determined by q-numbers,

and moving in opposite directions according to (38). For n=3 we have zeros determined by q-numbers,

x 2 (t) = 0, (42)

two of which are moving in opposite direction according to (38) and one is in the rest.

Following similar calculations as in Proposition I we have next relation e q νtD 2 x e q (kx) = e q (νtk 2 )e q (kx).

(44)

The right hand side of this expression is the plane wave type solution of the q-heat equation

Expanding both sides in power series in k we get q-Kampe de Ferie polynomial solutions of this equation

Consider an arbitrary analytic function f

is a time dependent solution of the q-heat equation ( 45).

According to this we have the evolution operator for the q-heat equation as

It allows us to solve the initial value problem

in the form φ(x, t) = e q νtD 2 x φ(x, 0 + ) = e

…(Full text truncated)…

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