Localization of Nonlocal Symmetries and Symmetry Reductions of Burgers Equation?

Jian-Wen Wu(吳劍文),Sen-Yue Lou(樓森岳),,? and Jun Yu(俞軍)2Center for Nonlinear Science and Department of Physics,Ningbo University,Ningbo 352,China

2Institute of Nonlinear Science,Shaoxing University,Shaoxing 312000,China

1 Introduction

Since Sophus Lie successfully solved differential equations by means of Lie group method[1]and Neother presented the affnity between symmetry and conservation law,[2]the Lie group approach has played the more and more important role in seeking solutions of differential equations.It is well known that the Lie group method is useful for obtaining explicit and particular solutions of a given equation according to solutions of ordinary differential equations.It provides us a way to construct new solutions with Lie point symmetry.One of us(Lou)indicated that the in finitely many nonlocal symmetries of integrable systems could be constructed by the inverse recursion operator,[3?4]the conformal invariance,[5]the inverse Lax operators,[6]the Darboux transformations,[7]the in finitely many Lax pairs,[8]the B?cklund transformations(BT),[9]the truncated Painlevé expansions,[10]and consistent tanh expansions.[11?12]In this paper,due to the truncated Painlevé expansion and the conformal transformation we obtain the nonlocal symmetries of the Burgers equation.According to the linear characters of symmetry equation,a linear superposition of several symmetries remains a symmetry,thus the multiple nonlocal symmetries are given.As we know nonlocal symmetries are diffcultly applied to construct exact solutions of differential equations.Hence,the localization of the multiple nonlocal symmetries is proposed.For(1+1)-dimensional integrable systems,to find the new solutions of solitons from symmetries,there are two kinds of effective methods.One of which is to find a finite transformation from the nonlocal symmetry.The other is to search for symmetry reductions.[13]Then by means of the symmetry reduction method,some forms of group invariant solutions including the interaction solutions among solitons and other nonlinear wave can be obtained.[14]

Recently,the Schwarzian Burgers equation and finite transformations of the multiple nonlocal symmetries of the Burgers equation are obtained by solving initial value problem.[15?16]The interaction solutions are given by using one nonlocal symmetry in Ref.[17].

The paper is organized as follows.In Sec.2,the multiple nonlocal symmetries are obtained with the truncated Painlevé method.In order to solve the initial value problem of nonlocal symmetries,one can transform the nonlocal symmetries of the original Burgers equation to the localized ones by introducing new dependent variables.In Sec.3,by solving the initial value problem of the localized symmetries for the enlarged system,we get the auto-B?cklund transformation of the Burgers equation.And in Sec.4,the Lie point symmetries and some types of symmetry reduction equations are obtained.In Sec.5,some interaction solutions are obtained by solving the symmetry reduction equations.The last section is a conclusion and discussion.

2 Schwarzian Burgers and Localization of Multiple Nonlocal Symmetries

The Burgers equation

whereνis the viscosity coeffcient. In order to find the nonlocal symmetries of the Burgers equation,the Schwarzian form of the Burgers equation can be introduced by the truncated Painlevé expansion.Based on theWTC method,the truncated Painlevé expansion reads[19]

wherefis an arbitrary singularity manifold.By balancing the nonlinear and dispersion terms of the Burgers equation,the corresponding truncated Painlevé expansion becomes

Substituting Eq.(3)into Eq.(1),we get

Vanishing the coeffcients off?3andf?2,we have

Collecting to the coeffcient off?1,f0,we get the following Schwarzian Burgers equation:

where

Then both

are solutions of Eq.(1),where Eq.(8)can be obtained from Eqs.(3)and(5).Thanks to the first equation of Eq.(5),we can find the nonlocal-symmetry related tou0.

Theorem 1(Multiple nonlocal symmetry theorem).Iffi,i=1,2,...,nare solutions of Eq.(6),the multiple nonlocal symmetries of the Burgers equation(1)can be read

Proof σis defined as a solution of the linearized system of Eq.(1)

which means Eq.(1)is form invariant under the transform

Due to Eq.(4)we find thatu1is a solution of Eq.(1)andu0is a solution of the symmetry equation by comparing the coeffcient off?1and the symmetry definition equation.Hence,we obtain a nonlocal symmetry which isfx～u0.In addition,fxandfare linked withunonlocally by Eq.(7).The multiple nonlocal-symmetries are obtained via the linear combinations ofnnonlocal symmetriesfi,i=1,2,...,n.The theorem 2 has been proved.

Because of the complexity of the multiple nonlocal symmetries,here,we only deal with a simple situation forn=2

Based on the standard Lie algebra theory,the finite transform is obtained by solving the initial value problem

It is diffcult to solve the initial value problem(13)due to the intrusion of the functionf1,f2and their differentiation.In order to solve the initial value problem related with the nonlocal symmetry(12),the nonlocal symmetry of Eq.(12)could transform to the Lie point symmetry in a suitable prolonged system.[15]By introducing new dependent variables,the prolonged system is given

The symmetriesσk(k=u,g1,g2,f1,f2)are the solutions of the linearized equations of the prolonged system(14)

it is easy to find the localized symmetries of the prolonged system(14)

wherec1andc2are arbitrary constants.

3 The Initial Value Problem

Thankstothelocalized nonlocalsymmetriesof Eq.(17),the initial value problem for the prolonged system is given

The auto-B?cklund transformation can be obtained by solving above initial value problem(18)

If{u,f1,f2,g1,g2}is a solution of the prolonged system(14),another solutionare obtained via above finite symmetry transformation.

4 Lie Point Symmetries and Symmetry Reductions

Besides finite transformation,symmetries can also be applied to get invariant solutions by reducing dimensions of a partial differential equations.The Lie point symmetries of the prolonged system Eq.(14)are invariant under the in finitesimal transformations

whereX,T,Σu,Σf1,Σf2,Σg1,and Σg2are functions ofx,t,u,f1,f2,g1,andg2.

Substituting the formula(21)into the symmetry definition equations(15)and collecting the coeffcients ofu,f1,f2,g1,g2,and their derivatives,we can obtain a set of determining equations for the in finitesimalsX,T,Σu,Σf1,Σf2,Σg1,and Σg2with the solution

whereci(i=1,2,3),x0,t0andcare arbitrary constants.To find the related symmetry reductions,we solve symmetry constraintsσk=0(k=u,f1,f2,g1,g2)defined by Eq.(21)with Eq.(22).It is equivalent to solve the corresponding characteristic equation[18]

There are four cases for the reduction equations.

In this case,by solving Eq.(23),the similarity solutions can be written as

with similarity variable

HereU,F1,F2,G1,andG2in Eq.(24)represent five group invariant functions.Substituting Eq.(24)into the prolonged system yields the symmetry reduction equation

It is obvious that once one gets the solutionU,F1,F2,G1,G2from Eq.(25),the exact solutions of the Burgers equation are obtained by Eqs.(24)and(25).

IfF1≡F1(η)is a solution of the reduction equation

then the solution for the Burgers equation reads

where

The similarity variable is

andc1,c0,C1,C2,x0,andt0are arbitrary constants.Case 3 c2t0=0,c1=0.

IfU≡U(τ)is a solution of the reduction equation

then the solution of Burgers equation is

with

The similarity variable isτ=(c0t2+2tx0? 2t0x)/2t0andc0,x0,t0,C1,C2,C3,C4are arbitrary constants.

Case 4c2t0=0,c1=0,c0=0.Similar procedure as above case,ifU≡U(?)is a solution of the reduction equation

withx0andt0are arbitrary constants and?=?(tx0?t0x)/t0is a similar variable.Then the solution of Burgers equation(1)reads

where

5 Interaction Solutions

In this section,we shall investigate the above reduction equations to construct some explicit interaction solutions of the Burgers equation Eq.(1).

Example 1(Soliton–Kummer waves). By solving Eq.(25),the exact solution for Burgers equation reads as

wherec,c0,c1,c2,c3,C1,C2,A,B,C,andDare arbitrary constants.The Kummer functions for KummerU(μ,ν,z)and KummerM(μ,ν,z)are the solution of the ordinary differential equation

It is obvious that the explicit solutions of Eq.(1)are immediately obtained via Eq.(35).The exact solutions can be considered as interaction solutions between the soliton and Kummer wave solution.

Example 2(Soliton-Airy waves).By solving Eq.(29),we obtain the solution

forc,c0,t0,x0,C5,andC6are arbitrary constants.Then substituting Eq.(37)into Eq.(38),the interaction solutionuis given

The AiryAi(μ)and AiryBi(μ)are the solutions of the ordinary differential equation

6 Conclusion and Discussion

In conclusion,the Schwarzian form of Burgers equation are obtained via the truncate Painlevé expansion.The multiple nonlocal symmetries are given according to the Schwarzian Burgers equation.In order to solve the initial value problem related to the multiple nonlocal symmetries,the Burgers equation is transformed to the prolonged system with the localization procedure.The auto-B?cklund transformation is given by solving the initial value problem of the prolonged systems.The interaction solutions for soliton-Kummer and the soliton-Airy waves are obtained with symmetry reductions related to multiple nonlocal symmetries.For the symmetry reduction related to multiple nonlocal symmetries,we study a situation of two nonlocal symmetries.It is interesting to further study exact solutions of nonlinear systems related to more nonlocal symmetries.

The authors is in debt to Drs.M.Jia,B.Ren,and X.Z.Liu for their helpful discussions.

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