Lihua Zhang, Bo Shen, Shuxin Han, Gangwei Wang and Lingshu Wang

School of Mathematics and Statistics,Hebei University of Economics and Business,Shijiazhuang 050061,China

Abstract In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated.The two complex nonlinear models are turned into multi-component real models by proper transformations.Lie symmetries are obtained via the classical Lie group method,and the results for the coupled complex short pulse equations contain the existing results as particular cases.Based on the linearizing operator and adjoint linearizing operator for the two real systems,adjoint symmetries can be obtained.Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair (SA) method.Relationships between the nonlinear selfadjointness method and the SA method are investigated.

Keywords: complex short pulse equation, coupled complex short pulse equations, symmetries,adjoint symmetries, conservation laws, SA method

1.Introduction

In the field of optical communications, nonlinear mathematical physical models that can describe the pulse propagation in optical fibers play an important role.The famous nonlinear Schr?dinger (NLS) equation [1, 2], the short pulse (SP)equation [3, 4], and the Wadati–Konno–Ichikawa equation[5,6]have been established respectively to describe different pulse propagations in optical fibers.Based on the SP equation, Feng proposed a new model.

withq=q(x,t)being a complex-valued function, and the equation was called complex short pulse (CSP) equation [7].Sinceq=q(x,t)comprises amplitude and phase, it is more accurate to describe optical waves.The equation (1) can describe the pulse propagation at the scale of attoseconds[8].The integrability of equation (1) has been studied in [7].N-soliton solutions, one-order rogue wave and soliton interaction patterns of equation (1) were investigated in [9].The relationship between the complex coupled dispersionless equations and equation (1) has been found [10].Periodic solutions, soliton solutions, higher-order breathers and rogue wave solutions for equation (1) have been obtained [11, 12].Long-time asymptotic behaviour and the general form of twosoliton solutions have been studied via Riemann–Hilbert approach [13, 14].More results about equation (1) can be seen in [15, 16].

When birefringence effects were considered,the coupled complex short pulse (CCSP) equations

were proposed [7], withq1andq2being complex-valued functions concerningxandt.Integrable properties, vector soliton solutions for equation (2) have been studied [7, 17].Lie symmetries and exact solutions of equation(2)have been obtained [18].

As we know, conservation laws is very fundamental not only in physics, but also in mathematics [19, 20].Many methods have been developed to find conservation laws[21–34].Recently, the adjoint equation has been applied to derive conservation laws in the nonlinear self-adjointness method and the symmetry/adjoint symmetry pair (SA)method [24, 25].To our best knowledge, explicit conservation laws of(1)and(2)have not been reported.This paper is concerned with the conservation laws of(1)and(2)by the SA method.

The framework of the rest is shown below.In section 2,we change equations (1) and (2) to real equations, then perform Lie symmetry analysis for the corresponding real equations.In section 3,conservation laws of the real systems is derived using the SA method.The last section is devoted to some conclusions and discussions.

2.Symmetry analysis of equations (1) and (2)

By taking

whereI2= -1 ,uandvare real dependent variables with respect toxandt, equation (1) is transformed to

In the following, we will investigate equation (4) instead of equation (1).By taking

Equation (2) are changed to

whereu,v,pandqare real dependent variables ofxandt.

As stated by Noether, Lie symmetries have close relationships with conservation laws.Next, we will study Lie symmetries for equations (4) and (5) by the classical Lie symmetry approach.Suppose that the Lie symmetry of equation (4) is expressed as follows:

where the coefficientsX,T,UandVare undetermined functions with respect tox,t,uandv.According to procedures of the classical Lie symmetry approach, (6) is determined by

andpr(2)Vdenotes the second prolongation ofV,

with

whereDxandDtare total differential operators.After calculations,the Lie symmetries for equation(4)can be obtained as follows

The Lie symmetries for equation (4) are new, and they can be used to discuss the corresponding symmetry groups,similarity reductions and group invariant solutions of equation (4).Similar work has been done in[32],so here we omit it.

By the same way with equation (4), we can get the Lie symmetries of equation (5)

In[18],equation(2)have been turned into the same real systems as equation(5).They also studied the Lie symmetries of equation (5), but their results are particular cases of (11).

3.Conservation laws of equations (4) and (5)

Based on the obtained Lie symmetries(10)and(11),we can construct conservation laws for equations(4)and(5)using their Lie symmetries and adjoint symmetries [25, 26].

3.1.Conservation laws of equation (4)

According to [25, 26], one obtains the linearizing operator of equation (4)

and the adjoint linearizing operator

Based on the symmetries (10) of equation (4), the symmetry componentsof equation (4) are given by

From (13), one obtains the following adjoint linearizing system

withω1andω2are functions ofx,t,uandv.After complicated calculations, we find that the solution to (15) is

whereCis a constant.According to[25,26],this is the adjoint symmetry of equation(4)and we takeC=1 in the calculation of conservation laws.

Using other pairs(i= 2, 3, 4)in (14) and (ω1,ω2) = (v, -u),one can obtain other three conservation laws of equation (4)

Remark 1.The validity of the above conservation laws has been checked by Maple.According to[33],there are two different types of trivial conservation laws.In the first type the divergence of the corresponding conserved vector vanishes identically.The conservation laws(X4,T4) belongs to the first type and is trivial, sinceDxX4+Dt T4≡ 0.The second type of trivial conservation laws are provided by those conserved vectors whose components vanish on the solutions of the considered systems of differential equations.The conservation laws(X1,T1)and(X2,T2)belongs to the second type and is also trivial.For example,

and

Therefore,(X1,T1)belongs to the second type of trivialness,so is(X2,T2).The conservation laws(X3,T3)is nontrivial and can be simplified to

By means of the above nontrivial conservation laws, equation (4) can be written in conservation form

The conservation form of (4) can help one find exact solutions of (4) [34].

Remark 2.From (15), we know that the adjoint equations of equation (4) are

Taking the solution (ω1,ω2) = (Cv,-Cu)into equation (21), we find that

According to the definition of quasi-self adjointness [29], we can conclude that equation (4) are quasi self-adjoint.

3.2.Conservation laws of equation (5)

According to [25, 26], one obtains the linearizing operator of equation (5)

and the adjoint linearizing operator

with

From (11), there are nine symmetry components of equation (5), they are given by

From (23), one obtains the following adjoint linearizing system

withωi(i= 1, 2, 3, 4)being functions ofx,t,u,v,pandq.The solution to (25) is

whereCis a constant.This is the adjoint symmetry of equation (5) and we takeC=1 in the following for simplicity.

Substituting other(i= 2, 3,K…,9)in (24) and(ω1,ω2,ω3,ω4)in (26), one can obtain other conservation laws of equation (5) with respect toVi(i= 2, 3,K…,9)

Remark 3.According to [33],(X4,T4) and(X8,T8) is trivial and belongs to the first type, since

(X1,T1) and(X2,T2) is trivial and belongs to the second type, for example

and

Other conservation laws for equation (5) is all nontrivial and(X3,T3) can be simplified to

Using the nontrivial conservation laws, equation (5) can be written in conservation form.

Remark 4.Similar to the case in Remark 2,we can directly obtain the adjoint equations of equation(5)from(25)and they are as follows

Taking the solution (ω1,ω2,ω3,ω4) = (-Cu,Cv, 0, 0)into equation (27), we find

According to the definition of quasi-self adjointness [29], we know equation (5) are quasi self-adjoint.

Remark 5.From the definition of adjoint symmetry, we can conclude that the determination of the self-adjointness and nonlinear self-adjointness (including quasi self-adjointness, weak self-adjointness, see [27, 30, 31]) in the nonlinear selfadjointness method can be determined by the adjoint symmetry in the SA method.

4.Conclusions and discussions

Recently,the CSP equation and CCSP equations have been proposed and attracted the interest of many researchers.The two equations can be regarded as the NLS equation and coupled NLS equations in the scale of attoseconds.In this paper,explicit conservation laws for equations(4)and(5)is obtained by the SA method.The relationships between nonlinear self-adjointness method and the SA method are investigated.The determination of the self-adjointness and nonlinear self-adjointness is very essential in the nonlinear self-adjointness method,and we find that they can be determined by the adjoint symmetry in the SA method.Also, we find that the conservation laws of equations (4) and (5) derived from SA method is all local.Using the derived nontrivial conservation laws, equations (4)and(5)can be written in conservation form.The conservation form can be employed to derive exact solutions and develop numerical and analytical methods [34].

Acknowledgments

This research was funded by the National Natural Science Foundation of China (No.12105073), Science and Technology Program of Colleges and Universities in Hebei Province(No.QN2020144), Science and Technology Plan Project(Special Program for Soft Science) in Hebei Province (No.20556201D), Scientific Research and Development Program Fund Project of Hebei University of Economics and Business(Nos.2020YB15, 2020YB12 and 2021ZD07), Youth Team Support Program of Hebei University of Economics and Business.