Zu-feng Liang,Xiao-yan Tang and Wei Ding

1 Department of Physics,Hangzhou Normal University,Hangzhou 10036,China

2 School of Mathematical Sciences,Shanghai Key Laboratory of PMMP,East China Normal University,Shanghai 200062,China

3 School of Computer Science and Technology,Shanghai Normal University,Shanghai 200234,China

Abstract Nonlocal symmetries related to the B?cklund transformation (BT) for the modified KdV-sine-Gordon(mKdV-SG)equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations.Then through the parameter expansion procedure,an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived.Finally,several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.

Keywords: Nonlocal conservation law,The modified KdV-sine-Gordon equation,Nonlocal symmetry

1.Introduction

Symmetries and conservation laws of nonlinear partial differential equations(NPDEs)have been studied intensively and extensively,because they play an important role in the investigation of integrable properties,invariant solutions,stability analysis,numerical calculations and so on.The conservation law and symmetry can also be used to construct methods to explore nonlocally related PDE systems which is important in the analysis of a given PDE system [1].The symmetry based method has been systematically shown to obtain non-invertible mappings of the Kolmogorov equation with variable coefficients to the backward heat equation,and the non-invertible mappings of linear hyperbolic PDEs with variable coefficients to linear hyperbolic PDEs with constant coefficients [2].Recently,generalized symmetries,cosymmetries and local conservation laws of the isothermal no-slip drift flux model have been exhaustively described in[3].The Lie symmetry analysis has also been used to study analytical solutions for time-fractional nonlinear systems such as the time-fractional Benjamin-Ono and Benjamin-Bona-Mahony equations with the Riemann-Liouville derivatives [4,5].

Lately,nonlocal symmetries associated with linearizing transformations,B?cklund transformations (BTs) and Darboux transformations have been studied a lot to construct highly nontrivial families of solutions,conservation laws and new integrable systems for some integrable systems [6-13].For instance,the residual symmetry can be derived by the truncated Painlevé method and thus used to construct n-th B?cklund transformations which lead to various solutions such as the lump and lump-type solutions [14,15].The nonlocal symmetries related to the BT have been studied for the sine-Gordon equation,and the associated topics on invariant solutions and nonlocal conservation laws have also been discussed [9,10].Here,we are concentrated on more nonlocal symmetries related to the BT for the modified KdVsine-Gordon (mKdV-SG) equation

and some related topics including the nonlocal conservation laws,considering its important application in physics.The mKdV-SG equation (1) was first proposed when exploring nonlinear wave propagation in a monoatomic lattice where the anharmonic potential effect competes with the dispersive one under the influence of weak dislocation potential [16].Then,in the study of optical pulse propagation in a medium described by a two-level Hamiltonian,it was rigorously demonstrated that the sG and mKdV equations can be derived from the Maxwell-Bloch equations by assuming the resonance frequency of the two-level atoms is either well above or well below the inverse of the characteristic duration of the pulse [17].In the presence of both high- and low-frequency resonances,the nonlinear propagation of ultrashort pulses can be well described by the mKdV-SG equation.The mKdV-SG equation has also been mathematically investigated including its integrability property [18] and various soliton solutions[19-24].

Mathematically,the mKdV-SG equation (1) was shown to describe pseudospherical surfaces,namely,it is the integrability condition for the structural equation of such surfaces,and by means of a geometrical method,the BT of the mKdV-SG equation was obtained as [25],

with

It means that if u is a solution of equation (1),then v determined by the BT(2)-(3)with(4)also satisfies the mKdV-SG equation in the form of

It is noted that a special nonlocal symmetry related to the BT(2)-(3) of equation (1) in its polynomial form has been obtained and some similarity solutions have been presented in [26].

The paper is arranged as follows.In the next section,three types of nonlocal symmetries of the mKdV-SG equation (1) associated with the BT (2)-(3) with (4) are obtained explicitly by requiring equations (1)-(4) form invariant under the infinitesimal transformations.Then infinitely many nonlocal conservation laws related to the nonlocal symmetries are obtained in section 3.Taking the nonlocal symmetries as symmetry constraint conditions applied on the BT,finite and infinite dimensional nonlinear systems are constructed in section 4.The last section is devoted to summary and discussions.

2.Nonlocal symmetries

Let us require the mKdV-SG equation (1) and its BT(2)-(3) are form invariant under the infinitesimal transformationsu→u+∈σu,v→v+∈σv,η→η+∈λ,where ∈is an infinitesimal parameter,σu,σvand λ are the symmetries ofu,vand η,respectively,then a linearized system can be derived as

It is shown that the symmetries σu,σvand λ satisfy a linear differential system of equations (6)-(9),however,it is still rather difficult to obtain its general solution.Therefore,we just write down three special solutions as below.

Case(1).The first type of nonlocal symmetry is obtained as

with p given by

It is easy to check thatpxt=ptxis satisfied identically.

Case(2).The second type of nonlocal symmetry is found to be

and λ is an arbitrary constant.Here,p satisfies equation(11),and q is determined by

withqxt=qtxsatisfied identically.

Case(3).The third type of nonlocal symmetry is given as

where p satisfies equation (11),and f is determined by

and the consistent conditionfxt=ftxis also satisfied identically.

It is remarkable that the above three special types of symmetries(10),(13)and(16)are called nonlocal because the function v in these symmetries is related to the function u through the BT (2)-(3).In addition,the first nonlocal symmetry(10)is equivalent to the nonlocal symmetry obtained in[26] for the mKdV-SG equation in its polynomial form.Likewise,we can make the special nonlocal symmetry (10)localized by introducing auxiliary functions and thus similarity solutions and similar interactive wave solutions can be obtained for the mKdV-SG equation (1).

Owing to the parameter η in these nonlocal symmetries σu,a series of infinitely many nonlocal symmetries can be generated straightforwardly.Here we would like to present a series of infinitely many nonlocal symmetries from the third nonlocal symmetry(16).Due to the procedure is routine as in[10,11]and can be programmed directly,we just write down the final results.Substituting the following expansions

where the expansion coefficientspi,vi,fiandare functions of x and t,and δ is an arbitrary expansion constant,into the third nonlocal symmetryin equation(16)together with equations (2),(3),(11),(12),(17),(18),and replacing η with η+δ,a series of infinitely many nonlocal symmetries can be obtained,where the first three are in the form of

where the functions vi,pi(i=0,1,2,3) and f0are determined by the following compatible conditions

3.Infinitely many nonlocal conservation laws related to the nonlocal symmetries

To find conservation laws for the mKdV-SG equation is nothing but to find the pairs of ρ and J satisfying

where ρ and J are called the conserved density and the conserved flux,respectively,for any solution u of equation (1).Many effective ways have been established to study conservation laws.Our previous work [10,11] show that the divergence expression (40) can be obtained for infinitely many conservation laws by applying the parameter expansion method either to some auxiliary functions involved in solving the symmetry equations (8) and (9),or directly to the BT.Here,we are only concentrated on infinitely many nonlocal conservation laws corresponding to the nonlocal symmetries related to the BT.

Integrating equations (8) and (9) with respect to x and t,respectively,gives

and

where the arbitrary integration functions have been simplified to a same constant C,p and q are determined by equations(11)-(12)and equations(14)-(15),respectively.To equal equations (41) and (42) arrives at

which leads to a nontrivial new nonlocal conservation law with the conserved density and flux as

and

respectively.

It is seen that the exponential part in this new nonlocal conservation law is related to the first nonlocal symmetrygiven by equation (10),while σvis any symmetry of the function v in equation (5) determined by the system of equations (6)-(9).Consequently,the nonlocal conservation law is related to the nonlocal symmetries.

Moreover,a series of infinitely many nonlocal conservation laws,in the form of0,i= 0,1,2,… ,can be obtained by applying the parameter expansion method,namely inserting the expansions of the functionsp,vin equation(19)into the conserved density(44),the flux(45)and the related equations(7)-(8),(11)-(12)with η replaced by η+δ,and then equalling zero the coefficients of the same orders of δ.For illustration and for simplicity,here we just write down the new nonlocal conserved density and flux for i = 1 as

and the corresponding nonlocal conserved flux reads

where v0,v1,p0and p1are determined by equations(23)-(26),(31)-(33),andare determined by

which are obtained in the same way by substituting the expansion of v in equation (19) andinto equation (7).

4.New integrable systems from nonlocal symmetries

In this section,we present some new nonlinear systems integrable in the sense of possessing infinitely many symmetries by means of the symmetry constraint method,namely,applying some nonlocal symmetry constraint conditions on the BT (2)-(3) of the mKdV-SG equation (1).

Let each pair (u,ui) ,(i=1,2,...,N) satisfy the BT,reading

with

Case 1: One dimensional nonlinear integrable systems.First we impose the generalized symmetry constraint with the first nonlocal symmetry (10)on the x-part of the BT(54) as

then the first kind of the finite dimensional(N+ 1)-component integro-differential equations can be obtained as

with arbitrary constants aiand ηi.Introduce

to simplify the symmetry constraint condition (57) as

and then to transform equations (58)-(59) into a nonlinear system of N-component ordinary differential equations in the form of

withwi=gix.It is noted that the above system is just the one obtained in [11],which demonstrates that from different original systems and nonlocal symmetries,the same nonlinear integrable system might be established via the symmetry constraint approach.

Second,we introduce the second nonlocal symmetry(13)on the x-part of the BT,namely

then equation (59) with the above constraint (63) constitutes the second kind of the finite dimensional(N+ 1)-component integro-differential equations.Under the same introduction(60),equation (63) can be rewritten as

with

Consequently,we have a nonlinear system of N-component integro-differential equations

with H given by (65).

Following the same way,other nonlocal symmetries can also be used to not only the x-part of the BT,but also to the tpart of the BT(55)to form new nonlinear integrable systems,but the results seem much more complicated.

Case 2: Higher dimensional nonlinear integrable systems.It is known that infinite dimensional nonlinear models can also be constructed in a similar way by introducing internal parameters,namely,imposing some internal parameter dependent symmetry constraints on the BT.

Let us take

as a new symmetry constraint condition,which is feasible because the mKdV-SG equation is invariant under the inner parameter y translation,and apply it on the x-part of the BT(2)to form a(1+1)-dimensional(N+ 1)-component integrodifferential system

with arbitrary constants aiandηi.The further application of the following transformation

on equations (68)-(69) arrives at a system of (1+1)-dimensional N-component differential equations

It is noted that the above system(71)is equivalent to the one presented in [11].In addition,the symmetry condition(67) and the others with other nonlocal symmetries can also be imposed on the t-part of the BT(55)with(56)to form new higher dimensional systems.For instance,applying the following nonlocal symmetry constraint

on the t-part of the BT (55) will lead to a (2+1)-dimensional N-component integro-differential equations,which are not given explicitly here for their complicated expressions.

5.Summary and discussions

In summary,the nonlocal symmetries and nonlocal conservation laws of the mKdV-SG equation are studied in detail.It is shown that the linearized equations of the mKdV-SG equation and its BT can give not only new nonlocal symmetries related to the BT,but also new nonlocal conservation laws related to the new nonlocal symmetries.In detail,three special nonlocal symmetries and one special conservation law are given explicitly.Then using the parameter expansion method,infinitely many nonlocal symmetries and infinitely many nonlocal conservation laws are constructed explicitly and straightforwardly.Finally,imposing symmetry constraints with the new nonlocal symmetries on the BT,finite and infinite dimensional systems of N coupled nonlinear equations are constructed,whose integrable properties need further considerations.As the mKdV-SG equation plays an important role in physics,it is really hoped that the results presented above might also find their applications in various physics.

Acknowledgments

The authors acknowledge the financial support by the National Natural Science Foundation of China (Grant Nos.11675055 and 12071302) and the Science and Technology Commission of Shanghai Municipality (Grant No.18dz2271000).