Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation

Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation Yong Chen ab and Xiaorui Hu b a Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai 0006 China b Nonlinear Science Center and Department of Mathematics Ningbo University Ningbo 5 China Reprint requests to Y C; E-mail: chenyong@nbueducn Z Naturforsch 6a 009; received April 00 / revised June 00 The classical symmetry method and the modified Clarkson and Kruskal C-K method are used to obtain the Lie symmetry group of a nonisospectral Kadomtsev-Petviashvili KP equation It is shown that the Lie symmetry group obtained via the traditional Lie approach is only a special case of the symmetry groups obtained by the modified C-K method The discrete group analysis is given to show the relations between the discrete group and parameters in the ansatz Furthermore the expressions of the exact finite transformation of the Lie groups via the modified C-K method are much simpler than those obtained via the standard approach Key words: Nonisospectral KP Equation; Classical Symmetry Method; Lie Symmetry Group; Modified C-K Method Introduction As it is well known the Kadomtsev-Petviashvili KP equation [] plays an important role in many fields of physics particularly in fluid mechanics plasma physics and gas dynamics David Kamran Levi and Winternitz [] studied the Lie point symmetry group of the KP equation via the traditional Lie group approach Lou and Ma [] developed the direct method presented by Clarkson and Kruskal C-K [] to construct the finite symmetry group of the KP equation However recently more and more people study the nonisospectral and variable coefficient generalizations of completely integrable nonlinear evolution equations In this paper we will investigate the nonisospectral KP equation [5 6] u t +yu xxx +6yuu x +xu y x +yu yy +u y = 0 which may provide more realistic models in the propagation of small-amplitude surface waves in straits or large channels of slowly varying depth and nonvanishing vorticity Here the classical symmetry method and the modified C-K method are used to obtain the Lie symmetry group of this nonisospectral KP equation Lie Symmetry Group of the Nonisospectral KP Equation Obtained by the Classical Symmetry Method In this section the classical symmetry method is used to obtain the Lie symmetry group of the nonisospectral KP equation Then some special solutions are given At first we give a brief outline of the theory of Lie s one-parameter group of transformations for invariance of a partial differential equation with three independent variables [] Generalization to more variables is straightforward We investigate a general partial differential equation with one dependent variable u and three independent variables x yandt: Hxytu x u y u t u xx u xy u xt u yy u yt u tt Let one parameter ε group of transformations of the variables x y t and u be taken as x = f xytu;ε t = pxytu;ε y = gxytu;ε u = hxytu;ε Let u = θxyt be a solution of If we replace the variables u x y t in by vx = f xytu;ε y = 09 0 / 09 / 000 000 $ 0600 c 009 Verlag der Zeitschrift für Naturforschung Tübingen http://znaturforschcom

Y Chen and X Hu Lie Symmetry Group and the KP Equation 9 gxytu;ε t = pxytu;ε respectively becomes Hx y t vv x v y v t v x x v x y v x t v y y v y t v t t =0 Then v = θx y t is a solution of That is to say in the case of transformation v = θx y t is a solution to whenever u = θxyt is a solution to This condition implies if and have a unique solution then θx y t =hxytθxyt;ε 5 Hence θxyt satisfies the one-parameter function equation θ f xytθ;εgxytθ;ε pxytθ;ε 6 = hxytθ;ε Expanding about the identity ε = 0 we can generate the following infinitesimal transformations: x = x + εξxytu+oε y = y + εηxytu+oε t = t + ετxytu+oε u = u + εφxytu+oε The functions ξ η τ φ are the infinitesimals of the transformations for the variables x y t and u respectively We shall denote the infinitesimals for u x u y u t u xx u xy u xt by φ x φ y φ t φ xx φ xy φ xt ; for example we have φ x =φ ξ u x ηu y τu t x + ξ u xx + ηu yx + τu tx = φ x + φ u u x ξ x + ξ u u x u x η x + η u u x u y τ x + τ u u x u t φ t =φ ξ u x ηu y τu t t + ξ u xt + ηu yt + τu tt = φ t + φ u u t ξ t + ξ u u t u x η t + η u u t u y τ t + τ u u t u t φ xx =φ ξ u x ηu y τu t xx + ξ u xxx + ηu yxx +τu txx φ xt =φ ξ u x ηu y τu t xt + ξ u xxt + ηu yxt + τu ttx φ yt =φ ξ u x ηu y τu t yt + ξ u xyt + ηu yyt + τu tyt Using these various extensions the infinitesimals criteria for the invariance of under the group is given by VH H=0 = 0 9 where the prolongation of the tangent vector field V is given by V = ξ x + η y + τ t + φ u + φ x u x + φ y 0 + φ t + φ xx + u y u t u xx Here and in the following we denote all ξ xytu ηxytu τxytu φxytu by ξ η τ φ The purpose is to solve ξ η τ φ by taking and 0 into 9 Then we collect together the coefficients of u u x u y u xx u xy u yy and set all of them to zero At last we get a system of linear partial differential equations from which we can find ξ η τ and φ in practice Next we will use the above method to find the Lie symmetry group of the nonisospectral KP equation The prolongation of the tangent vector field of is V = ξ x + η y + τ t + φ u + φ x u x + φ y + φ xx + φ xy + φ xt u y u xx u xy + φ yy + φ xxxx u yy u xxxx u xt Then 9 reads [here H =u t + yu xxx + 6yuu x + xu y x + yu yy + u y ]: φ xt + yφ xxxx + yu x φ x + 6yuφ xx + 6yu xx φ + 6φ y + xφ xy + yφ yy + u xxxx η + 6u x η + 6uu xx η + u xy ξ + u yy η = 0 Substituting and u xxxx = y u xt + 6yu x + 6yuu xx + 6u y + xu xy + yu yy for φ x φ y φ xx φ xy φ xt φ yy φ xxxx in we get a system of linear partial differential equations We set all the coefficients of u x u y u t u xx u yy u xt to zero Then we get ξ = y g t t+y ht+xf t t+ xgt y yf tt t η = yf t t+y gt τ = f t 5 φ = xht uf t t ugt 9 y 5 y xg t t 9 y f tttt g t t y + h t t y x 6 gt y

0 Y Chen and X Hu Lie Symmetry Group and the KP Equation So the infinitesimal operator of is Q = ξ x + η y + τ t + φ u = y g t t+y ht+xf t t+ + yf t t+y gt y + f t t xht + uf 9 y 5 t t ugt y xgt y + h t t f tttt g t t x gt y y u = xf t t yf tt t x + yf tt y + f t t + f tttt uf t t u + + xgt y g tt t y + y ht x + y y g t t x + y gt y x gt y h t t y yf tt t x xg t t 9 y xg t t ugt 9 y y u + xht 9 u = X f +Yg+Zh Here f t gt and ht are all arbitrary functions of t Q is a general element of the Lie algebra of The commutation relations for this Lie algebra are easily to obtain: y 5 [X f X f ] = X f ḟ ḟ f [X f Y g] = Y f ġ ḟg [X f Zh] = Z f ḣ ḟh [Y g Y g ] = Zġ g g ġ [Y gzh] = 0 [Zh Zh ] = 0 where the dots indicate derivatives with respect to tto solve from we can also get d dε x y t u =Qx y t u x y t u ε=0 =xytu or dx ξ = dy η = dt τ = du φ = dε 9 Here ξ η τ φ are obtained by insteading x y t u in ξ η τ φ by x y t u From 9 5 and 6 we can get x y t u but their impressions are too complicated For simplicity we only discuss the case where f t =gt =0ht 0 From we obtain Q = Zht = y ht x + h t t + xht y 9 y 5 u 0 Then we solve the equations dy 0 = dt = dε 0 dx dε = y ht du dε = h t t + x ht y 9 y 5 The solution is u = h t ε + h t t+ htx ε y y y + ux y htεyt That is to say if uxyt is a solution of u in is also a solution of Lie Symmetry Group of the Nonisospectral KP Equation Obtained by the Modified C-K Method In this section we will use another method a simple direct method to investigate In [] Clarkson and Kruskal introduced a direct method to derive symmetry reductions of a nonlinear system without using any group theory For many types of nonlinear systems the method can be used to find all possible similarity reductions In [] Lou and Ma modified the C-K direct method to find the generalized Lie and non-lie symmetry groups for the well-known KP equation Here we will use this modified direct method to discuss At first we introduce the main idea of this modified C-K direct method [ 0] Given a general partial differential equation PDE with the variables x i and u Fx i uu xi u xi x j =0 i j = n 5 The general form of solutions of 5 reads u = Wx x x n UX X X n X i = X i x x x n i = n 6

Y Chen and X Hu Lie Symmetry Group and the KP Equation It requires U to satisfy the same PDE 5 under the transformation ux x x n UX X X n We can see that it is enough to suppose that the group transformation has some simple forms say ux x x n =αx x x n +β x x x n UX X X n instead of 6 for various nonlinear systems especially for those models where the C-K direct reduction method does work Next we will apply this method with the proof of the generality of to At first we will show that in the following ξ η τ are functions of x y t and that they are different from ξ η τ in Section There are no relations between them Substituting u = WxytUξ ητ into and requiring Uξ ητ is also a solution of the nonisospectral KP equation but with independent variables Uξ ητ [eliminating U ξτ and its higherorder derivatives by means of ] Then we get Using condition is reduced to yw UU ξ x U ξξ + G xytuu ξ =0 Now vanishing the coefficient of U ξξ in we have W UU = 0 5 That is to say we can use u = αxyt+β xytuξ ητ 6 to substitute the general solution Now the substitution of 6 with into the nonisospectral KP equation leads to βξ x yξ x τ t ηu ξξξξ + ξ x yβ x ξ x + βξ xx U ξξξ +6yββ xx + β x U + 6βξ x yβξ x τ t ηu ξ + [ 6yβ β x ξ x + yξ xx U ξ + 6βξ x yβξ x τ t ηu ξξ +xβ xy + yβ xxxx + 6yαβ xx + 6yβα xx + yβ yy + 6β y +β xt +yα x β x ] U+ yβξyy + yβ x ξ xxx + yαβ x ξ x +6yβ xx ξ xx + β t ξ x + 6yβ y ξ y + yβα x ξ x + 6βξ y +yξ x β xxx +6yαβξ xx yβξ xxxx + xβ y ξ x + β x ξ t +xβ x ξ y + βξ xt + xβξ xt Uξ +β x η t + 6βη y y W U ξ x τ x U ξ 0 +GxytUU ξ U ξ 9U η U τ =0 9 +yβη yy 6βξ x τ t + 6yβ y η y + xβ x η y U η + β x τ t U τ + yβξ xx +6yαβξ x +βξ x ξ t +yβξ y +yβξ x ξ xxx where U ξ n = n U and G is a complicated U ξ n ξ 0- independent function Equation 9 is true for an arbitrary solution U only for all coefficients of the polynomials of the derivatives of U being zero From 9 the coefficient of U ξ 0 should be zero So we have y W U ξ x τ x = 0 0 Obviously W U should not be zero and one can prove that there is no nontrivial solution for ξ x = 0; so the only possible case of 0 is τ x = 0 i e τ = τyt Substituting into 9 we get yw U τ y U ττ +η x U η +G xytuu ξ =0 Setting the coefficients of U ττ and U η in to zero we have τ τt η ηyt +yβ x ξ x ξ xx + 6yβ xx ξ x + xβξ x ξ y Uξξ + β ξ x η t ξξ x τ t + xξ x η y + yξ y η y Uξη + β ξ x τ t η +yη y U ηη + xα xy + yα yy + 6yαα xx + yα xxxx +α xt + 6yα x + 6α y = 0 Setting the coefficients of the polynomials of U and its derivatives to be zero can be read as yξ x τ t η = 0 β x ξ x + βξ xx = 0 ββ xx + β x = 0 β x ξ x + yξ xx = 0 xβ xy + yβ xxxx + 6yαβ xx + 6yβα xx + yβ yy +6β y + β xt + yα x β x = 0 yβξ yy + yβ x ξ xxx + yαβ x ξ x + 6yβ xx ξ xx 9 +β t ξ x + 6yβ y ξ y + yβα x ξ x + 6βξ y 0 +yξ x β xxx + 6yαβξ xx yβξ xxxx + xβ y ξ x + β x ξ t +xβ x ξ y + βξ xt + xβξ xy = 0

Y Chen and X Hu Lie Symmetry Group and the KP Equation β x η t + 6βη y + 6yβ y η y + yβη yy 6βξ x τ t + xβ x η y = 0 β x τ t = 0 yβξ x τ t η = 0 yβξ xx + 6yαβξ x + βξ x ξ t + yβξ y + yβξ x ξ xxx + yβ x ξ x ξ xx + 6yβ xx ξ x + xβξ x ξ y = 0 ξ x η t ξξ x τ t + xξ x η y + yξ y η y = 0 ξ x τ t η + yη y = 0 xα xy + yα yy + 6yαα xx + yα xxxx + α xt + 6yα x + 6α y = 0 The result reads 5 ξ = [τ t δc τ t y η0 + c η 0 y + yτ t x y τ t τ tt δc τ t τ t η 0t + τ tt η 0 y 5 τ t c η 0 τ tt + 6c τ t η 0 η 0t δτ t ξ0 y + 6τt η 0 δη 0 η 0t + c τ t ξ0 y 5 +τ t η0 c η 0 η 0t + δc τ t ξ0 y ] / τ t y δy τt + c η 0 η = δc y τt + η 0 β = τt δτ t + c η 0 6 α = τ t η0 + y y 5 η0 + 6δc τ t η0 y + 9c τ t y η 0 + 6δτ t y 5 0 η0 + δc τ t y + c τ t y [c δτ t τ t η 0t τ tt η 0 y 0 + 6τ t δτ t ξ0 + 6c τ t η 0 η 0t 6c τ tt η 0 y +τ t 0 η0 δτ tt η 0 + δτ t η 0t η 0 c τ t ξ0 y τ t η0 c η 0 τ tt + 9δc τ t ξ0 + 6c τ t η 0 η 0t y τ t η 0 6δc η 0 τ tt + 9δc τ t η 0 η 0t 6τ t ξ0 τ t η0 δc τ t ξ0 + 6τ t η 0 η 0t + η 0 τ tt y 5 +δc τ t τt τ ttt η 0 6τ tt η 0 τ t η 0t τ tt + 9τ t η 0tt y y ]x + τ t τt τ ttt τ tt y +6τ t δτ t τtt ξ 0 c τ t τ tt η 0t η 0 + 0c τ t η 0tt η 0 + 0c τ t τ ttt η 0 c τ tt η 0 δτ t ξ0t δτ t η 0 t y +τ t 696δτ t τ tt η 0t η 0 + 9c τ t τtt η 0 ξ 0 + 0δτ t η 0 τ ttt 6c τ t η0 ξ 0t 656δη 0 τ tt + 6δτ t η 0 η 0tt 0δτ t η 0t η 0 + 6c τ t η0t ξ 0 y +τ t 5 c η 0 τ tt + 60c τ t η 0 τ ttt + 0δc τ t η0t ξ 0 η 0 50δc τ t η0 ξ 0t 9τ t ξ0 +60c τ t η 0 η 0tt 5c τ t η 0 η 0t + δc τ t τtt η 0 ξ 0 c τ t η 0 η 0t τ tt y 0 +τ t η 0 δc τ t η 0 τ ttt 6δc τ t ξ0 0τ t η0 ξ 0t 60δac τ t η 0 η 0tt +δc τ t τ ttt η 0 η 0t + τ t η0t ξ 0 η 0 06δc τ t η 0 η 0t 9δc η 0 τ tt +5τ t η0 τ tt ξ 0 y +τ t η0 η 0 τ tt + δc τ t η0 τ tt ξ 0 5c τ t ξ0 50δc τ t η0 ξ 0t +6τ t η 0 τ tt η 0t δc τ t η0 η 0t ξ 0 5τ t η 0 η 0t + τ t η 0 τ ttt 6τ t η 0 η 0tt y +6τ t η0 c τ t η0 τ tt ξ 0 c τ t ξ0 η 0 η 0t δτ t 5 ξ0 + 0δc η 0 τ tt η 0t 6c τ t η0 ξ 0t 0δc τ t η 0 η 0tt 0δc τ t η 0 η 0t y x

Y Chen and X Hu Lie Symmetry Group and the KP Equation +η 0 6δτ t η0 ξ 0t + δτ t η0 τ tt ξ 0 c τ t η 0 η 0t + c η 0 τ tt η 0t τ t y 5 6δτ t η0 ξ 0 η 0t 9c τ t ξ0 c τ t η 0 η tt y / δc y τt η0 + c η 0 + τ t y δy τt + c η 0 Here ξ 0 ξ 0 t η 0 η 0 t τ τt are arbitrary functions of time t while the constants δ and c possess discrete values determined by δ = c = + i i i = At last we get if U = Uxyt is a solution of the nonisospectral KP equation u = α + τ t δτ t + c η 0 y Uξ ητ with 5 6 and where ξ 0 η 0 τ are arbitrary functions of t From with 5 6 and we know that for the real nonisospectral KP equation the symmetry group is divided into two sections: the Lie point symmetry group which corresponds to δ = c = and a coset of the Lie group which is related to δ = c = The coset is equivalent to the reflected transformation of x and yiey yt t accompanied by the usual Lie point symmetry transformation We denote by S the Lie point symmetry group of the real nonisospectral KP equation NKP by σ the reflection of y yt tby I the identity transformation and by C Iσ the discrete reflection group Then the full Lie symmetry group G RNKP of the real nonisospectral KP equation can be expressed as G RNKP = C S For the complex nonisospectral KP equation the symmetry group is divided into six sectors which correspond to δ = c = δ = c = + i δ = c = i δ = c = δ = c = + i δ = c = i That is to say the full symmetry group G CNKP expressed by with 5 6 and for the complex nonisospectral KP equation is the product of the usual Lie point symmetry group S δ = c = and the discrete group D ie G CNKP = D S D Iσ yt R R σ yt R σ yt R where I is the identity transformation and σ yt : yt y t R : uxyt + i u R : uxyt + i u + i + i x + i x + i y + i t y + i t We will show that the Lie symmetry group obtained via the traditional Lie approach is only a special case of the symmetry groups obtained by the modified C-K method When δ = c = we can find the Lie point symmetry group from with 5 6 and and we can see its equivalence with the result obtained in Section We set τt=t +ε f t ξ 0 t= εht η 0t= εgt 9 with an infinitesimal parameter ε Then with 5 6 and with δ = c = can be written as u = U + εσu+oε

Y Chen and X Hu Lie Symmetry Group and the KP Equation σu= xf t t+ xgt y g t t yf tt t+y ht U x + yf t t+gty U y + f tu t y y gt + + f t t U + f tttt+ x gt + g tt t y y + xg t t 9 y 9 50 xht h t t y 5 y The equivalent vector expression of the above symmetry is V = xf t t yf tt t x + yf tt y + f t t + Uf tt f ttt U + y g t t+ xgt y x + y gt y + Ugt g tt x gt xg t t y y y 9 y U + y ht x + h t t + xht = X f t +Ygt + Zht 9 U y y 5 5 We can see that 5 is exactly the same as which we have obtained by the standard Lie approach in Section Remark: In [] Lou and Ma have obtained the full symmetry group of the complex KP equation It can be divided into six sectors In our result the full symmetry group of the complex nonisospectral KP equation is also be gotten similarly to the KP equation and it also has six sectors It is very interesting that the full Lie symmetry groups of the isospectral and nonisospectral KP equation have the similar algebraic structure Conclusions Based on two methods: the classical symmetry method and a simple direct method two Lie symmetry groups of a nonisospectral KP equation have been constructed It has been shown that the Lie symmetry group obtained via the traditional Lie approach is only a special case of the full symmetry groups obtained by the modified C-K method Using the modified C-K method we can also get more solutions of the equation from the old ones At the same time we showed that the full symmetry groups of the isospectral and nonisospectral KP equation have the same algebraic structure However from our result it was shown that the nonisospectral problem is much more complicated than the isospectral one Its calculation is more complex and corresponding reduction is more difficult than the isospectral one That might be because of its inherent variable spectral parameter which also makes the equation more valuable and general Acknowledgements We would like to thank Prof Senyue Lou for his enthusiastic guidance and helpful discussions The work is supported by the National Natural Science Foundation of China Grant No 0500 and Grant No 900 Shanghai Leading Academic Discipline Project No B K C Wong Magna Fund of Ningbo University and Program for Changjiang Scholars and Innovative Research Team in University IRT0 [] B B Kadomtsev and V I Petviashvili Sov Phys Dokl 5 59 90 [] D David N Kamran D Levi and P Winternitz J Math Phys 5 96 [] S Y Lou and H C Ma J Phys A: Math Gen 9 005 [] P A Clarkson and M Kruskal J Math Phys 0 0 99 [5] D Y Chen H W Xing and D J Zhang Chaos Solitons and Fractals 5 6 00 [6] S F Deng and Z Y Qin Phys Lett A 5 6 006 [] M Lakshmanan and P Kaliappan J Math Phys 95 9 [] H C Ma and S Y Lou Commun Theor Phys 6 005 006 [9] H C Ma Chin Phys Lett 55 005 [0] H C Ma Commun Theor Phys 0 005