On Burgers equation on a time space scale

R E S E A R C H

Open Access

On Burgers equation on a time-space

scale

Gro Hovhannisyan

_{, Logan Bonecutter and Anita Mizer}

*_{Correspondence:} ghovhann@kent.edu

Kent State University at Stark, 6000 Frank Ave NW, North Canton, OH 44720, USA

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.

MSC: 35Q51; 35K05; 34N05

Keywords: Burgers equation; heat equation; partial differential equations; time scales; spectral expansion; Fourier series; soliton-like nonlinear equations; Lax equation

1 Introduction

In  Lax [] introduced the linear operator equation equivalent to the nonlinear Korteweg-de Vries (KdV) equation that describes the traveling solitary waves. The im-portance 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. Lax equation may be used to generate other nonlinear dynamic equations with the properties mentioned above.

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 ex-tension of AKNS method on difference equations. Other nonlinear dynamic equations are studied in [–].

In [] Hilger introduced the time scale calculus that unifies continuous and discrete analysis.

The papers [–] are the first articles dedicated to KdV-like dynamic equations on time scales. In [] the notion of regular-discrete time scale was introduced (see Section ). Also in [] some KdV-like equations on a regular-discrete space scale were deduced by using the Gelfand-Dickey method.

LetTandXbe arbitrary nonempty closed subsets of real numbers. The setsT,Xare called the time and space scales correspondingly. The setT×X={(t,x),t∈T,x∈X}we call the time-space scale.

In [] soliton-like equations on a regular-discrete time-space scale were obtained by using the AKNSL method.

In this paper we extend the Lax matrix equation on a time-space scale dynamic system. From this equation we deduce the viscous Burgers (see (.) below) and the nonlinear Schrodinger dynamic equations on a time-space scale by using the AKNSL method.

We expect that this extension will give a wider range of integrable nonlinear dynamic equations that could be used in modeling.

We show that the Burgers equation on a time-space scale can be linearized by using the Cole-Hopf transformation. We also derive the formulas for solutions of the boundary value problem for the Burgers equation (see (.) and the heat equation on a time-space scale. These formulas are pretty simple and may be used to study the wave motion on a time-space scale.

2 Basic notations from the time scale calculus

Fort∈Tandx∈X, we define backward jump operatorsσ(t) :T→T,ρ(x) :X→X

σ(t) :=sup{s∈T:s<t}, ρ(x) :=sup{y∈X:y<x}. (.)

Forx∈X, we define the forward jump operatorβ(x) :X→Xby

β(x) :=inf{y∈X:y>x}. (.)

The graininess functionsμ(t) :T→[,∞),ν(x),α(x) :X→[,∞) are defined as

μ(t) =t–σ(t), ν(x) =x–ρ(x), α(x) =β(x) –x. (.)

We are considering nabla time and space derivatives [] instead of delta derivatives [] since in physics the applications of nabla derivatives by time variable are casual.

IfThas a right-scattered minimumm, defineTκ:=T–{m}; otherwise, setTκ=T. Forv:T→Randt∈Tκ, define the nabla derivative [] ofvatt, denoted byv∇t(t), to be the number (provided it exists) with the property that given anyε> , there is a neighborhoodUof t such that|v(ρ(t)) –v(s) –v∇t_{(t)[ρ(t) –s]}| ≤_ε|_{ρ(t) –s}|_{for alls}∈_U.

ForT=R, we havev∇t_{(t) =v}

t(t), the usual derivative, and forT=Z, we have the backward

difference operatorv∇t_{(t) :=v(t) –v(t– ).}

In the same way one can define the nabla derivativev∇x_{(x) by space (x) variable.}

We are going to use the following denotations:

w(t)(t,x) :=w∇t_{(t,x) =lim} s→t

w(s) –w(σ(t))

s–σ(t) , (.)

w(x)(t,x) =w∇x_{(t,x) :=w}∇x_{(x) =lim} y→x

w(y) –w(ρ(x))

y–ρ(x) , (.)

where

In view of the assumption thatμ(t) depends only on time variable andν(x) depends only on space variable, we have

μ(x)(t) =ν(t)(x) = , fσρ(t,x) =fρσ(t,x) :=fσ(t),ρ(x).

Lemma . If the functions f(t,x),μ(t),ν(x)are twice nabla differentiable,then

f(tx)(t,x) =f(xt)(t,x), fρt(t,x) =ftρ(t,x). (.)

For the proof of this lemma, see [].

Frequently we are going to use the product and the quotient rules (see [])

f(t,x)g(t,x)(t)=f(t)(t,x)g(t,x) +fσ_(t,x)g(t)_(t,x),

f(t,x)

g(t,x) (t)

=f

(t)_{(t,x)g(t,x) –f(t,x)g}(t)_(t,x) gσ_(t,x)g(t,x) ,

f(t,x)g(t,x)(x)=f(x)(t,x)g(t,x) +fρ(t,x)g(x)(t,x),

f(t,x)

g(t,x) (x)

=f

(x)_{(t,x)g(t,x) –f(t,x)g}(x)_(t,x) gρ_(t,x)g(t,x) .

(.)

We say that a functionθ(·,x) :T→Rist-regressive providedθ(·,x) is ld-continuous and  –μ(t)θ(t,x)=  holds for allt∈Tand allx∈X. We say that a functionθ(t,·) :X→Ris

x-regressive providedθ(t,·) is ld-continuous and  –ν(x)θ(t,x)=  holds for allt∈Tand allx∈X.

Ifθ(t,x) ist-regressive, the nablat-exponential functioneˆθ(t,t,x) on a time scaleTcan

be defined as the unique solution of the initial value problem (see [, ])

ˆ

e(_θt)(t,t,x) =θ(t,x)eˆθ(t,t,x), eˆθ(t,x,x) = . (.)

Ifθ(t,x) isx-regressive, the nablax-exponential functionˆeθ(t,x,x) on a space scaleXis defined similarly as the unique solution of the initial value problem

ˆ

e(θx)(t,x,x) =θ(t,x)eˆθ(t,x,x), eˆθ(t,x,x) = . (.)

Note that

fρ(t,x) =f(t,x) –ν(x)f(x)(t,x), fσ(t,x) =f(t,x) –μ(t)v(t)(t,x). (.)

3 Lax equation

Consider the nabla dynamic systems

v(x)(t,x) =M(t,x)v(t,x), M(t,x) =

M(t,x,z) M(t,x)

M(t,x) M(t,x,z)

, (.)

v(t)_{(t,x) =N(t,x)v(t,x), N(t,x) =}

A(t,x) B(t,x)

C(t,x) D(t,x)

wherev(t,x) =v(t,x)

v(t,x)

andMkj(t,x,z),k,j= , , are the functions that may depend on the

spectral parameterzas well.

We derive the Lax equation as the compatibility condition of two linear dynamic systems (.), (.).

From (.), (.) by differentiation and usage of the product rule we get

v(xt)=M(t)v+Mσ_v(t)_=M(t)_v+Mσ_Nv,

v(tx)=N(x)v+Nρ_v(x)_=N(x)_v+Nρ_Mv.

Here and further we often suppress (t,x) or (t,x,z) variables to shorten the formulas. By equating the mixed derivativesv(xt)=v(tx), we get the Lax matrix equation

N(x)(t,x) +Nρ(t,x)M(t,x,z) =M(t)(t,x,z) +Mσ(t,x,z)N(t,x), (.)

or in component form

A(x)+AρM+BρM=M_(t)+M_σA+M_σC,

C(x)+CρM+DρM=M_(t)+Mσ_A+M_σ C,

B(x)+Aρ_M+Bρ_M=M(t)

+MσB+MσD, D(x)+Cρ_M

+DρM=M(t)+MσB+Mσ D.

(.)

In view ofAρ_=A–ν(x)A(x)_,Aσ _=A–μ(t)A(t)_{, we get}

A(x) –ν(x)M+BρM–Mσ_C=M(_t) –μ(t)A,

D(x) –ν(x)M+Cρ_M–Mσ

B=M (t) 

 –μ(t)D,

C(x) –ν(x)M

+CM–Mσ

+Dρ_–AM

=M(t)

 –μ(t)A,

B(x) –ν(x)M

+BM–Mσ

+Aρ_–DM

=M(t)

 –μ(t)D.

Choosing

M(t,x,z) =Q(t,x), M(t,x,z) =R(t,x),

we have

A(x) –ν(x)M+BρR–QσC=M(_t) –μ(t)A,

D(x) –ν(x)M+CρQ–RσB=M(_t) –μ(t)D,

C(x) –ν(x)M+CM–Mσ_+Dρ–AR=R(t) –μ(t)A,

B(x) –ν(x)M+BM–Mσ



+Aρ_–DQ=Q(t)_{ –μ(t)D.}

(.)

In a continuous time scaleμ(t) = ,Mσ_{(t,x) =M(t,x) system (.) is simplified as} fol-lows:

A(x)( –νM) =Mt+QC–BρR,

D(x)( –νM) =Mt+RB–QCρ,

C(M–M) =RA–Dρ–C(x)( –νM) +R(t),

B(M–M) =B(x)( –νM) –QD–Aρ–Q(t).

(.)

If both time scale and space scale are continuous, that is,T=R,X=R,ν(x) =μ(t)≡, system (.) is further simplified, and we get the system of Lax equations introduced in []:

Ax=Mt+QC–BR, Dx=Mt+RB–QC,

C(M–M) =R(A–D) –Cx+Rt, B(M–M) =Q(A–D) +Bx–Qt.

4 Burgers equation on a time-space scale

The viscous Burgers equation occurs in mathematical models of gas dynamics, traffic flow, the flow through a shock wave traveling in a viscous fluid, and some probabilistic models [, ].

To derive a time-space scale version of the Burgers equation, consider the Lax equation (.) in the scalar case (A=A,B=C=D=R=Q=M≡)

A(_x)(t,x) –ν(x)M(t,x,z)

=M(_t)(t,x,z) –μ(t)A(t,x)

. (.)

By choosing

M(t,x,z) =F(t,x)z+F(t,x),

we get

A(_x)(t,x) –ν(x)F(t,x)z–ν(x)F(t,x)

=F_(t)(t,x)z+F_(t)(t,x) –μ(t)A(t,x), (.)

or, assuming thatzis an arbitrary spectral parameter that does not depend on timet, we get

A(_x) –ν(x)F=F_(t) –μ(t)A, –A(_x)ν(x)F=F_(t) –μ(t)A,

or

ν(x)A(_x)

( –μ(t)A)=

ν(x)F_(t)

 –νF = – F_(t)

F , F (t)

 ( –νF) +νF (t)  F= .

Using the quotient differentiation rule on a time scale

F F

(t)

=F

(t)  F–F

(t)  F Fσ

F

we get

F_(t)+ν(x)F_(t)F–F(t)F

= , F

(t)  Fσ

F

+ν(x)

F F

(t)

= ,

ν(x)F F –



F

(t)

= , ν(x)F– 

F =C= constant,

F_(t)=  –ν(x)F  –μ(t)AA

(x)  .

Choosing

F(t,x) :=F(t,x)

p , A= A p,

A(t,x) :=F(t,x) +p–ν(x)F(t,x)F(x)(t,x),

(.)

we get the Burgers equation on a time-space scale

F(t)(t,x) = p

_{–ν(x)F(t,x)} p_{–μ(t)A(t,x)}A

(x)_{(t,x), (.)}

whereA(t,x) is given in (.) andp_{is a viscosity constant coefficient.}

In a continuous time scaleT=R,μ(t)≡, we get the equation

Ft(t,x) =

 –νF(t,x)/pF(t,x) +p–ν(x)F(t,x)F(x)(t,x)(x). (.)

Note that the Burgers equation (.) on a space scale first was introduced in [].

If both time scale and space scale are continuous, that is,T=R,X=R,ν(x) =μ(t)≡, (.) turns to the classical Burgers equation []

Ft(t,x) =

F(t,x) +pFx(t,x)

x. (.)

To linearize (.), consider the Cole-Hopf transformation

F(t,x) =p

_v(x)_(t,x)

v(t,x) . (.)

By using the quotient rule we have

F(x)=v

(xx)_v–v(x)_v(x) vρ_v p

_{, F}(t)₌v(xt)v–v(x)v(t) vσ_v p

_,

 –ν(x)

p F=  –ν(x) vx

v = vρ

v,

A=F+p–ν(x)FF(x)=(p

_v(x)₎ v +

p_vρ

v ·

v(xx)_v–v(x)_v(x) vρ_v =

p_v(xx) v ,

and if

then

A=p

_v(xx) v =

p_v(t)

v ,  – μ

pA=  –μ v(t)

v = vσ

v,

and in view of

vσ

v(x) v

(t)

=vρ

v(t) v

(x)

,

equation (.)

F(t)=v ρ

vσA

(x)

turns to the identity

p_v(x) v

(t)

=v ρ

vσ

p_v(t) v

(x)

.

So ifv(t,x) satisfies the heat equation (.), then F(t,x) =p_vv₍(x_t)_,x(t₎,x) satisfies the Burgers equation (.).

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 solve (.) by looking for a solution in a product form (the separation or Fourier method [])

v(t,x) =T(t)X(x).

By the substitution in (.) and the separation of variables, we get

T(t)_(t) T(t) =

p_X(xx)_(x) X(x) = –λ

_{= const,}

T(t) =ˆe–λ(t,t), X(x) =K(λ)eˆ_iλ/p(x,x) +K(λ)ˆe–iλ/p(x,x),

whereK(λ),K(λ) are independent of (t,x),eˆ–λ(t,t) andˆe±iλ/p(x,x) are the nabla

expo-nential functions onTandXcorrespondingly.

Thusv(t,x,λ) =eˆ–λ(t,t)K(λ)eˆ_iλ/p(x,x) satisfies (.).

Since equation (.) is linear, it satisfies the superposition principle, that is, a linear com-bination of solutions is a solution (see []).

By the superposition principle the solution of (.) may be written in the form (see [])

v(t,x) = ∞

–∞

K(λ)eˆ–λ(t,t)ˆe_iλ/p(x,x)dλ. (.)

From the initial conditionv(t,x) =ϕ(x) we get

ϕ(x) = ∞

–∞K(

So the formal solution of the initial value problem (.) is given by (.), whereK(λ) is the solution of the integral equation (.).

Consider the initial value problem for Burgers equation

F(t)(t,x) = p

_{–ν(x)F(t,x)} p_{–μ(t)A(t,x)}A

(x)_{(t,x), F(t,x) =f(x), (.)}

whereA(t,x) =F_{(t,x) + (p}_{–ν(x)F(t,x))F}(x)_(t,x).

From the initial condition

f(x) =F(t,x) =p

_v(x)_(t ,x) v(t,x) =

p_ϕ(x)_(x) ϕ(x)

we get

ϕ(x) =ˆef/p(x,x), x∈X.

By substitution (.) into (.) we obtain the representation of solutions of the Burgers equation on a time-space scale

F(t,x) =

∞

–∞iλpK(λ)ˆe–λ(t,t)eˆ_iλ/p(x,x)dλ

∞

–∞K(λ)eˆ–λ(t,t)ˆeiλ(x,x)dλ

, (.)

whereK(λ) may be found by inversion of the Fourier transformation

ˆ

ef/p(x,x) =

∞

–∞

K(λ)eˆiλ/p(x,x)dλ. (.)

Note that the inversion of a Fourier transformation on some time scales was studied in [, ], but there is no inversion formula for an arbitrary time (space) scale.

Consider the boundary value problem

v(t,x) =ϕ(x), v(x)(t,x) =v(x)(t,b) = , x,b,x∈X,t,t∈T, (.)

for heat equation (.) on a time-space scale.

Introducing Bohner-Peterson’s trigonometric functions on a space scaleX(see [, , ])

sinλ/p(x,x) = ˆ

eiλ/p(x,x) –ˆe–iλ/p(x,x)

i ,

cosλ/p(x,x) = ˆ

eiλ/p(x,x) +eˆ–iλ/p(x,x)

 ,

and using the method of separation of variablesv(t,x) =T(t)X(x), one can rewrite the solutions ofp_Xxx_+λ_{X(x) =  in the form}

X(x) =Csinλ/p(x,x) +Ccosλ/p(x,x), p,λ∈R. (.)

From the boundary conditions (.) we get

In the caseν(x) =  the boundary condition turns to

and the solution of (.), (.) is given by the classical Fourier series formula

v(t,x) =

Note that using L’Hospital’s rule one can prove

we get

ˆ

eiλk/p(b,x) ˆ

e–iλk/p(b,x)

=ˆeiλk/(p+iλkν)(b,x) =eˆa(b,x) = ,

and

sinλk/p(b,x) = ˆ

eiλk/p(b,x) –eˆ–iλk/p(b,x)

i = ,

b

x

ˆ

ea(x,x)∇x=ˆ ea(x,x)

a

b

x

=ˆea(b,x) – 

a = .

Remark . To obtain the formula for coefficientsBkon an arbitrary space scale similar

to the remarkable formula (.) on a continuous space scale, it would be interesting to extend the orthogonality of the exponential functions on the unit circle to Hilger expo-nential functions, but we do not think it is possible.

Indeed, ifM=mπ ν

b–x, we have on an arbitrary space scale (ν(x) > )



b–x b

x

ˆ

e_(–ei(K–M)_)/ν(x,x_)∇x=δ_km=

⎧ ⎨ ⎩

, k=m,

, k=m,

but this property does not imply the orthogonality of Hilger exponents as we have for usual exponential functionseikπ(x–x)/(b–x)_,e–imπ(x–x)/(b–x)_:



b–x b

x

eikπ(x–x)/(b–x)_e–imπ(x–x)/(b–x)_dx=δ

km, k,m∈Z.

So the second condition (.) is satisfied if the eigenvaluesλkare chosen as in (.),

and the formal solution of the boundary value problem (.), (.) is given by Fourier series formula with Bohner-Peterson’s trigonometric functions

v(t,x) = ∞

k=

Bkcosλ_k/p(x,x)eˆ–λ_k(t,t), (.)

andBkcould be found from the initial condition

ϕ(x) = ∞

k=

Bkcosλk/p(x,x). (.)

Consider the boundary value problem

F(t,x) =f(x), F(t,x) =F(t,b) = , t,t∈T,x,x∈X (.)

for the Burgers equation (.).

By substitution (.) into (.) we get the representation of solutions of boundary value problem (.), (.)

F(t,x) = –p

∞

k=λkBksinλk/p(x,x)eˆ–λ_k(t,t)

∞

k=Bkcosλk/p(x,x)eˆ–λ_k(t,t)

where the numbersBkmay be found from the initial condition

ˆ

ef/p(x,x) =

∞

k=

Bkcosλk/p(x,x), (.)

since by takingt=tin (.), we get

f(x) =p

_ϕ(x)_(x)

ϕ(x) , ϕ(x) =eˆf/p(x,x).

Example . Consider the caseX=hZ. From (.) we get

λ=λk= – p htan

khπ b–x

, k= , , , . . . ,N.

Sinceb–x=Nh, whereNis some natural number, we have the finite numbers of different λk, that is, ≤k<N.

By takingx=xm,m= , , . . . ,N– , we get the linear system with respect to unknown

numbersBk

ϕ(xm) = N–

k=

Bkcosλk/p(xm,x), m= , , , . . . ,N– .

Remark . The formal formulas (.), (.), (.), (.) have sense if the Fourier integrals or series are convergent. In the caseX=nZthe sums in (.) and (.) are finite and there is no need to prove the convergence.

It would be interesting to figure out for which class of functionsϕ(x) the series (.), (.) are convergent for an arbitrary space scale. This could be a topic of a separate paper. Note that to prove the convergence similar to the classical harmonic analysis in an arbi-trary space scale, one needs to use the following properties:

() boundedness of the functionscos(λk(x–x)/p),eˆ–λ_k(t,t). () limk→∞λk=∞.

() The space scale analogue of formula (.).

Note that property () may be proved, but property () is not true since from (.) lim_k→∞λkdoes not exist in the caseν(x) > .

To compare with the classical case, consider the continuous space scaleX=R. In this case ifϕ(x)∈L(R) andϕ(x) is continuous onR, (.) turns to the usual Fourier transfor-mation, and using the inverse transformation we get from (.)

K(λ) =  π

∞

–∞

ϕ(y)e–ipλ(y–x)_dy= 

π

∞

–∞ e



p

y

x_f(x)dx–ipλ(y–x)

dy. (.)

So in the continuous space scale by substitution (.) into (.) we get the following formula for the solution of (.):

v(t,x) =  π

∞

–∞

ϕ(y) ∞

–∞e

iλ

p(x–y)_eˆ

and the formula for a solution of (.) be further simplified (we choose heret= ) as follows:

v(t,x) = 

we get the Poisson formula for a solution of the initial value problem (.)

v(t,x) = 

Furthermore, one can get the well-known formula for the solution of initial value problem for the classical viscous Burgers equation

F(t,x) =p ∂

5 Nonlinear Schrodinger equation on a space scale

Choose in (.)

M(t,x,z) =F(t,x)z+F(t,x), M(t,x,z) =G(t,x)z+G(t,x), z(t)= ,

Mt(t,x,z) =Ft(t,x)z+Ft(t,x), Mt(t,x,z) =Gt(t,x) +zGt(t,x),

(.)

and consider the spectral expansion

A(t,x) =A(t,x)z+A(t,x)z+A(t,x), B(t,x) =B(t,x)z+B(t,x),

C(t,x) =C(t,x)z+C(t,x), D(t,x) =D(t,x)z+D(t,x)z+D(t,x).

(.)

Denoting

˜

Aj=Aj(t,x) –Dρj(t,x), D˜j=Dj(t,x) –Aρj(t,x), j= , , ,

˜

Fj=Fj(t,x) –Gj(t,x), j= , ,

we get from (.)

Az+Az+A

(x)

( –νFz–νF)

=F_(t)z+F_(t) –μAz–μAz–μA

+Qσ_(C

z+C) –R(Bz+B)ρ,

Dz+Dz+D(x)( –νGz–νG)

=G(_t)z+G(_t) –μDz–μDz–μD+Rσ(Bz+B) –Q(Cz+C)ρ,

(Cz+C)(x)( –νFz–νF) + (Cz+C)F–Gσ_z+F–Gσ_

=R(t) –μAz–μAz–μA–RDρ_–Az+Dρ_–Az+Dρ_–A,

(Bz+B)(x)( –νGz–νG) + (Bz+B)G–F_σz+G–F_σ

=Q(t) –μDz–μDz–μD–QAρ_–Dz+Aρ_ –Dz+Aρ_–D.

By equating the terms next to the powers of parameterzk_{,k= , , , , we get}

–νFA(_x)= –μAF_(t),

( –νF)A(_x)–νFA_(x)= –μAF_(t)–μAF_(t),

( –νG)D(_x)–νGD_(x)= –μDG(_t)–μDG(_t),

–νFC(_x)+CF–Gσ_+Dρ_–AR= –μAR(t),

–νGB(_x)+BG–F_σ+Aρ_–DQ= –μDQ(t),

( –νF)A(_x)–νFA(_x)= ( –μA)F_(t)–μAF_(t)+QσC–RBρ_,

( –νG)D(_x)–νGD(_x)= ( –μD)G_(t)–μDG(_t)+RσB–QCρ_,

( –νF)C_(x)–νFC_(x)+F–Gσ_C+F–Gσ_C=A–Dρ_R–μAR(t),

( –νG)B(_x)–νGB_(x)+G–F_σB+G–F_σB=D–Aρ_Q–μDQ(t),

( –νG)D(_x)= ( –μD)G_(t)+RσB–QCρ_,

( –μA)R(t)= ( –νF)C_(x)+CF–Gσ



+RDρ_–A, (.)

( –μD)Q(t)= ( –νG)B(_x)+BG–F_σ+QAρ_–D. (.)

Considering the continuous time scale caseT=R,μ(t)≡ and choosing

F(t,x) = , G(t,x) = , F(t,x) =G(t,x), (.)

we get

A_(x)=D(_x)=A_(x)=D(_x)≡, (.)

C_ρ=RA˜, B=QA˜, νA(x)=RB

ρ

 –QC, RB–QC

ρ

 = , (.)

C_ρ=RA˜ – ( –νF)C_x, B=QA˜ + ( –νF)Bx_=QA˜ +A˜ ( –νF)Qx, (.)

A(_x)=F

(t)

 +QC–RB

ρ



 –νF , D

(x)  =

F_(t)+RB–QC_ρ

 –νF , (.)

Rt= ( –νF)C(x)+R

Dρ_–A

, Qt= ( –νF)B(x)+Q

Aρ_–D

. (.)

In the caseν(x) =  from (.)-(.) we have

C=RA˜ , B=QA˜ , C=RA˜ –RxA˜ , B=QA˜ +QxA˜

A= –RQA˜ , D= –A=RQA˜

and the evolution equations (.) on a space scale

Rt=Cx– RA, Qt=Bx+ QA

turn to the nonlinear system for unknown potentials (see [])R(t,x),Q(t,x)

Rt=RxA˜–A˜

Rxx– RQ

, Qt=QxA˜+A˜

Qxx– QR

, t∈R,x∈R. (.)

Considering the caseν(x) > , from (.) we get

C_(x)(t,x) =C(t,x)

ν(x) –

R(t,x)A˜

ν(x) , C(t,x) = –A˜

x

x

ˆ e/ν

x,ρ(y)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.

Further from (.) we get

C_ρ(t,x) =C(t,x) –ν(x)C_x(t,x) =RA˜ – ( –νF)C–RA˜

ν , (.)

RB–QC

ρ

=R

QA˜+A˜( –Fν)Qx

–Q

RA˜– ( –νF)

C–RA˜ ν

=A˜ ( –Fν)RQx–Q/ν+ ( –Fν)QC

For the consistency of this expression and (.), we assume thatFsatisfies

By direct calculations from (.)

Since (.) is the first order linear partial integro-differential equation with variable co-efficients with respect toF(t,x), one may prove the existence of a solution by using the

linear theory.

Evolution nonlinear equations (.) on a space scale

Rt(t,x) = ( –νF)Cx–R

under the assumption that the solutionF(t,x) of (.) exists, become

Rt(t,x) = ( –νF)C(x)– RA+νA

6 Nonlinear Schrodinger equation on a regular-discrete space scale

A space scaleXis called regular-discrete [] if the following two conditions are satisfied simultaneously:

ρβ(x)=x, βρ(x)=x for allx∈X,

whereρandβare the backward and forward jump operators, respectively. From this def-initionρis invertible for the regular-discrete space scaleXandρ–=β.

Setx∗=minXif there exists a finiteminX, andx∗= –∞otherwise. Also setx∗=maxX

if there exists a finitemaxX, andx∗= –∞otherwise.

Lemma .[] A space scaleXis regular-discrete if and only if the following two conditions hold:

() The pointx∗is right-dense and the pointx∗is left-dense.

() Each point ofX–{x∗,x∗}is either two-sided dense or two-sided scattered.

In particular,R,hZ, andKqare regular-discrete space scales.

Let us derive a nonlinear Schrodinger equation on a space scale assuming that the space scaleXis regular-discrete.

Note that the system of evolution nonlinear equations obtained in [] has sense only on the regular-discrete space scales.

From (.)-(.) we have

C=RβA˜ , B=QA˜ , νA(x)=RB

ρ

 –QC= –νA˜

RβQ(x),

C_ρ=RA˜ – ( –νF)C_x,

(.)

or

C_ρ=RA˜ –A˜ ( –νF)Rβ(x), B=QA˜ + ( –νF)QxA˜ . (.)

Since

RB–QC_ρ= ( –νF)A˜ RQx+QRβ(x)_{= ( –νF)A}˜ _QRβ(x)_,

QC–RBρ_=QC_ρ+νCx_–RB–νBx_=νQC_x+RBx_– ( –νF)A˜ QRβ(x),

from (.), assuming (.) is true, we get (.) and

A(_x)=F

(t)  +ν(QC

(x)  +RB

(x)  )

 –νF –A˜

QRβ(x), A_(x)= –A˜ QRβ(x).

Further from (.) we get the linear partial differential equation with respect to unknown functionF(t,x):

Ft(t,x) =νA˜

QRβ(x)β_{( –νF}

)β

(x)

–RQ(x)( –νF)

(x)

–νA˜

Rβ(x)_Q+Q(x)_R,

or

Ft(t,x) =ν

_˜

In view of (.)

and we get the following nonlinear system of dynamic equations on a space scale:

Rt(t,x) = ( –νF)

In a special caseA˜ = , we get more simple equations

under the assumption

Re[A˜ ] =Re[A˜ ] =Im[A˜ ] = , Q(t,x) = –R(t,x).

Furthermore, ifA˜ = ,A˜ = –i, equation (.) turns to the nonlinear Schrodinger equa-tion

iQt(t,x) =Q(xx)(t,x) –Q(t,x)Q(t,x). (.)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The idea to deduce and solve the Burgers equation on a time-space scale by using the AKNSL method, the Cole-Hopf transformation, and the separation of variables belongs to Gro Hovhannisyan. The proofs are verified by all authors. All authors read and approved the final manuscript.

Acknowledgements

Authors would like to thank anonymous reviewers for useful comments that helped to improve the original manuscript.

Received: 3 June 2015 Accepted: 26 August 2015

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