Fractional Breaking Soliton Equation Reduced from a Linear Spectral Problem Associated with Fractional Self-Dual Yang-Mills Equations

ZHANG Sheng MA Lina XU Bo

1 School of Mathematical Sciences,Bohai University,Jinzhou 121013,China 2 School of Mathematics,China University of Mining and Technology,Xuzhou 221116,China 3 School of Educational Sciences,Bohai University,Jinzhou 121013,China

Abstract:Fractional or fractal calculus is everywhere and very important. It is reported that the fractal approach is suitable for insight into the effect of porous structure on thermo-properties of cloth. A novel local fractional breaking soliton equation is derived from the reduction of the linear spectral problem associated with the local fractional non-isospectral self-dual Yang-Mills equations. More specifically,the employed linear spectral problem is first reduced to the (2+1)-dimensional local fractional zero-curvature equation through variable transformations. Based on the reduced local fractional zero-curvature equation,the fractional breaking soliton equation is then constructed by the method of undetermined coefficients. This paper shows that some other local fractional models can be obtained by generalizing the existing methods of generating nonlinear partial differential equations with integer orders.

Key words:fractional calculus;local fractional breaking soliton equation;local fractional non-isospectral self-dual Yang-Mills equations;(2+1)-dimensional local fractional zero-curvature equation

Introduction

Born more than 300 years ago,fractional calculus has important applications in many fields[1].Fractional calculus plays an irreplaceable role in textile engineering,especially in the study of heat transfer characteristics of textiles like those in Refs.[2-4].Fractional calculus has its unique advantages over traditional integral calculus.For example,when the classical continuum mechanics is invalid for the carbon nanotube because of its discontinuity,the fractional calculus is considered as an optional tool to deal with some problems in hierarchical or porous media[5-7].It is worth mentioning that Ain and He[8]propose a new definition of the two-scale dimension instead of the fractal dimension for dealing with most of discontinuous problems.For the same phenomenon,different results or laws can be reached by means of different scales.When the effect of the thickness is ignored,a piece of cloth can be considered as a two-dimensional object.Then one has to adopt the fractal approach[9]to studying the effect of the porous structure on thermo-properties of cloth.

With the development of fractional calculus,more and more attention has been paid to not only the solutions[10-18]but also the derivation of fractional order models[17-19].Solving or constructing fractional differential equations has theoretical and practical value,and has become a hot topic in the research field,because they can help us to understand the essence behind the phenomenon.Some analytical methods are valid to solve fractional differential equations,such as the exp-function methods[20],variational iteration methods[21],and homotopy perturbation methods[22].In order to derive the fractional order models,one can consider the methods for the derivation of their corresponding partners,namely nonlinear partial differential equations with integer orders.Researchers[16-18]focused on the dynamic characteristics of fractional solitons,especially those that showed different properties from conventional solitons in the evolution process.It was found in Refs.[17-18] that the propagation velocity of a fractional soliton was affected by the fractional order.An interesting evolutionary feature reported by Zhangetal.[18]is that the smaller the value of the fractional order is,the faster the soliton of the time-fractional KdV equation propagates at the initial stage.However,after a short time,the situation is just the opposite.The propagation of small fractional order soliton is slower than that of large fractional order soliton.

Recently,with the help of the derived formula,Zhangetal.[19]derived the local fractional non-isospectral self-dual Yang-Mills equations:

(1)

where

(2)

(3)

Note that the local fractional partial derivative[1]of orderα(0<α≤1) ofu(x,y,t) with respect toxis defined:

(4)

where Δα[u(x,y,t)-u(x0,y,t)]?Γ(1+α)[u(x,y,t)-u(x0,y,t)] with Γ(·) denoting the Gamma function.The local fractional partial derivatives have many important properties,some of which will be used in this paper:

(5)

(6)

(7)

(8)

wherekandqare arbitrary constants.

1 Fractional Zero-Curvature Equation

Lemma1If

(9)

then

(10)

(11)

ProofWe begin with the proof of the first sub-equation of Eq.(10).Letφbe an arbitrary local fractional differentiable function with respect tox1,x2,x3andx4.A direct computation gives

(12)

from which we have

(13)

namely the first sub-equation of Eq.(10).For the second sub-equation of Eq.(10),we have

(14)

and obtain

(15)

which can be written as the second sub-equation of Eq.(10).By the similar way,we get

(16)

(17)

and obtain

(18)

which can be respectively written as the two sub-equations of Eq.(11).

Lemma2Suppose thatu(x,y,t) andλ(y,t) are two functions of the indicated variables.Consider the known local fractional linear non-isospectral problem[19]:

(19)

(20)

which are associated with the local fractional non-isospectral self-dual Yang-Mills Eq.(1).Here the non-isospectral parameterλsatisfies the local fractional nonlinear partial differential equations of the forms:

(21)

Let Eqs.(14) and (15) be independent ofu,

(22)

(23)

Then Eqs.(19) and (20) can be reduced to the (2+1)-dimensional local fractional zero-curvature equation:

(24)

ProofUsing Eqs.(9) and (22),we have

(25)

In view of Eq.(2),from Eqs.(19) and (20) we have

(26)

(27)

With the help of Eqs.(23) and (25),from Eqs.(26) and (27) we have

(28)

At the same time,it is easy to see that Eq.(21) can be written as

(29)

Differentiating Eq.(28),we have

(30)

(31)

(32)

2 Fractional Breaking Soliton Equation

Theorem1Let

(33)

Then the local fractional breaking soliton Eq.(3) can be derived from the (2+1)-dimensional local fractional zero-curvature,i.e.Eq.(24).

ProofWe suppose that

(34)

wherea,b,c,d,gandhare all undetermined constants.Substituting Eq.(34) into Eq.(24) yields

(35)

We further seta=2,b=-1/2,c=2,d=-1,g=-1/4 andh=1.Then Eq.(35) is reduced to Eq.(24).

3 Conclusions

Since Lemma 2 tells that the (2+1)-dimensional local fractional zero-curvature,i.e.Eq.(24)，can be derived from the reduction of the linear spectral problem associated with the local fractional non-isospectral self-dual Yang-Mills,i.e.Eq.(1),we can conclude that the local fractional breaking soliton,i.e.Eq.(3)，also can be reduced from the related linear spectral problem by using Theorem 1.The soliton dynamical evolution of Eq.(3) is worthy of study.