3.1. Two Modification Rules and Formulas of Conservation Laws for the ANNV Equation
To search for conservation laws of (1) by Theorem 3, Lie symmetry, formal Lagrangian, and adjoint equation of (1) must be known. According to Definition 1, the adjoint equation of (1) is
(14)vyt+vyxxx-6uxyvx-3uyvxx-3uxvxy=0,
where v is a new dependent variable with respect to x, y, and t.
According to Theorem 2, the formal Lagrangian for the system consisting of (1) and (14) is
(15)L=(uyt+uyxxx-3uxxuy-3uxuxy)v,
where v is a solution of (14).
Suppose that the Lie symmetry for the ANNV equation (1) is as follows:
(16)V=ξ∂∂x+η∂∂y+τ∂∂t+ϕ∂∂u.
From Theorem 3, we get the general formula of conservation laws for the system consisting of (1) and (14):
(17)X=ξL+W((∂L∂uxxxy)∂L∂ux-Dx(∂L∂uxx)-Dy(∂L∂uxy) -Dxxy(∂L∂uxxxy))+Dx(W)(∂L∂uxx+Dxy(∂L∂uxxxy))+Dy(W)(∂L∂uxy+Dxx(∂L∂uxxxy))+Dxx(W)(-Dy(∂L∂uxxxy))+Dxy(W)(-Dx(∂L∂uxxxy))+Dxxy(W)(∂L∂uxxxy),Y=ηL+W((∂L∂uxxxy)∂L∂uy-Dx(∂L∂uxy)-Dt(∂L∂uyt) -Dxxx(∂L∂uxxxy)) +Dx(W)(∂L∂uxy+Dxx(∂L∂uxxxy))+Dt(W)(∂L∂uyt)+Dxx(W)(-Dx(∂L∂uxxxy))+Dxxx(W)(∂L∂uxxxy),T=τL+W(-Dy(∂L∂uyt))+Dy(W)(∂L∂uyt),
where W is the Lie characteristic function, W=ϕ-ξux-ηuy-τut, and L is the formal Lagrangian determined by (15).
In fact, because of the existence of the mixed derivative terms uxy, uyt, and uxxxy, the general formula of conservation laws must be modified; otherwise the previous X, Y, and T do not satisfy
(18)(DxX+DyY+DtT)|uxxxy=3uxxuy+3uxuxy-uyt=0.
The rules of modifications are as follows.
(
1
)
In one conservation vector (X, Y, or T), the time that one derivative with respect to a mixed derivative term appears is determined by the order of the derivative with respect to its independent variables. For example, whether in X or in Y, ∂L/∂uxy can only appear once; ∂L/∂uxxxy can only appear once in Y and can appear three times in X; ∂L/∂uxxxt can only appear once in T and can appear three times in X; ∂L/∂uxxt can only appear once in T and can appear two times in X.
(
2
)
The location that one derivative with respect to a mixed derivative term appears at cannot be the same in different conservation vectors. That is to say, if there is W(-Dy(∂L/∂uxy)) in X, then the term appears in Y can only be Dx(W)(∂L/∂uxy) and the term W(-Dx(∂L/∂uxy)) cannot appear in Y at the same time. And if there is W(-Dxxx(∂L/∂uxxxy)) in Y, then the terms that appear in X contain Dxxy(w)(∂L/∂uxxxy) and first and second total derivatives of ∂L/∂uxxxy.
Applying the two rules to the general conservation laws formula in Theorem 3, we can get the following results.
Theorem 4.
Suppose that the Lie symmetry of the ANNV equation (1) is expressed as (16). According to the different locations of ∂L/∂uxy, ∂L/∂uyt, and ∂L/∂uxxxy, the symmetry provides sixteen different conservation laws for the system consisting of (1) and (14). The conserved vectors are given as follows:
(19)(Xij,Yij,Tij)=(Xi,Yi,Ti)+(BjX,BjY,0), i,j=1,2,3,4,
with
(20)X1=ξL+W(∂L∂ux-Dx(∂L∂uxx))+Dx(W)(∂L∂uxx)+W(-Dy(∂L∂uxy)),Y1=ηL+W(∂L∂uy-Dt(∂L∂uyt))+Dx(W)(∂L∂uxy),T1=τL+Dy(W)(∂L∂uyt),X2=ξL+W(∂L∂ux-Dx(∂L∂uxx))+Dx(W)(∂L∂uxx)+W(-Dy(∂L∂uxy)),Y2=ηL+W(∂L∂uy)+Dx(W)(∂L∂uxy)+Dt(W)(∂L∂uyt),T2=τL+W(-Dy(∂L∂uyt)),X3=ξL+W(∂L∂ux-Dx(∂L∂uxx))+Dx(W)(∂L∂uxx)+Dy(W)(∂L∂uxy),Y3=ηL+W(∂L∂uy-Dx(∂L∂uxy)-Dt(∂L∂uyt)),T3=τL+Dy(W)(∂L∂uyt),X4=ξL+W(∂L∂ux-Dx(∂L∂uxx))+Dx(W)(∂L∂uxx)+Dy(W)(∂L∂uxy),Y4=ηL+W(∂L∂uy-Dx(∂L∂uxy))+Dt(W)(∂L∂uyt),T4=τL+W(-Dy(∂L∂uyt)),B1X=Dxxy(W)(∂L∂uxxxy)+Dxy(W)(-Dx(∂L∂uxxxy))+Dy(W)(Dxx(∂L∂uxxxy)),B1Y=W(-Dxxx(∂L∂uxxxy)),B2X=W(-Dxxy(∂L∂uxxxy))+Dxxy(W)(∂L∂uxxxy)+Dxy(W)(-Dx(∂L∂uxxxy)),B2Y=Dx(W)(Dxx(∂L∂uxxxy)),B3X=W(-Dxxy(∂L∂uxxxy))+Dxxy(W)(∂L∂uxxxy)+Dx(W)(Dxy(∂L∂uxxxy)),B3Y=Dxx(W)(-Dx(∂L∂uxxxy)),B4X=W(-Dxxy(∂L∂uxxxy))+Dxx(W)(-Dy(∂L∂uxxxy))+Dx(W)(Dxy(∂L∂uxxxy)),B4Y=Dxxx(W)(∂L∂uxxxy),
where W is the Lie characteristic function and W=ϕ-ξux-ηuy-τut, L is the formal Lagrangian determined by (15).
3.2. Explicit Conservation Laws of the ANNV Equation
Now, conservation laws of (1) can be derived by Theorem 4 if Lie symmetries of (1) are known. In fact, Lie symmetries of (1) have been obtained in [19] and they are as follows:
(21)V1=g(y)∂∂y, V2=-F(t)∂∂u,V3=f(t)∂∂x-xft3∂∂u,V4=xht3∂∂x+h(t)∂∂t-(ht3u+htt18x2)∂∂u,
where g(y), F(t), f(t), and h(t) are arbitrary functions.
Using the Lie symmetry V1 and Theorem 4, we can get sixteen conservation laws for the system consisting of (1) and (14). They are listed as follows:
(22)X111=-3g(y)uy2vx-3g(y)uyuxvy-vgyuxxy-vg(y)uxxyy-vxxgyuy-vxxg(y)uyy+vxgyuxy+vxg(y)uxyy,Y111=g(y)vuty+g(y)vuxxxy+g(y)uyvt+g(y)uyvxxx,T111=-vgyuy-vg(y)uyy,X112=-3g(y)uy2vx-3g(y)uyuxvy-vgyuxxy-vg(y)uxxyy+vxxyg(y)uy+vxgyuxy+g(y)vxuxyy,Y112=g(y)vuty+g(y)vuxxxy+g(y)uyvt-g(y)uxyvxx,T112=-vgyuy-vg(y)uyy,X113=-3g(y)uy2vx-3g(y)uyuxvy-vgyuxxy-vg(y)uxxyy+vxxyg(y)uy-g(y)uxyvxy,Y113=g(y)vuty+g(y)vuxxxy+g(y)uyvt+g(y)uxxyvx,T113=-vgyuy-vg(y)uyy,X114=-3g(y)uy2vx-3g(y)uyuxvy+vxxyg(y)uy+g(y)uxxyvy-g(y)uxyvxy,Y114=g(y)vuty+g(y)uyvt,T114=-vgyuy-vg(y)uyy,X211=-3g(y)uy2vx-3g(y)uyuxvy-vgyuxxy-vg(y)uxxyy-vxxgyuy-vxxg(y)uyy+vxgyuxy+vxg(y)uxyy,Y211=g(y)vuxxxy+g(y)uyvxxx,T211=g(y)uyvy,X221=-3g(y)uy2vx-3g(y)uyuxvy-vgyuxxy-vg(y)uxxyy+g(y)uyvxxy+vxgyuxy+vxg(y)uxyy,Y221=g(y)vuxxxy-g(y)uxyvxx,T221=g(y)uyvy,X231=-3g(y)uy2vx-3g(y)uyuxvy-vgyuxxy-vg(y)uxxyy+g(y)uyvxxy-g(y)uxyvxy,Y231=g(y)vuxxxy+g(y)uxxyvx,T231=g(y)uyvy,X241=-3g(y)uy2vx-3g(y)uyuxvy+g(y)uyvxxy+g(y)uxxyvy-g(y)uxyvxy,Y241=0,T241=g(y)uyvy,X311=-3g(y)uy2vx+3g(y)uxyuyv+3uxvgyuy+3uxvg(y)uyy-vgyuxxy-vg(y)uxxyy-vxxgyuy-vxxg(y)uyy+vxgyuxy+vxg(y)uxyy,Y311=g(y)vuty+g(y)vuxxxy-3g(y)vuxxuy-3g(y)vuxuxy-3g(y)uyuxvx+g(y)uyvt+g(y)uyvxxx,T311=-vgyuy-vg(y)uyy,X321=-3g(y)uy2vx+3g(y)uxyuyv+3uxvgyuy+3uxvg(y)uyy-vgyuxxy-vg(y)uxxyy+g(y)uyvxxy+vxgyuxy+vxg(y)uxyy,Y321=g(y)vuty+g(y)vuxxxy-3g(y)vuxxuy-3g(y)vuxuxy-3g(y)uyuxvx+g(y)uyvt-g(y)uxyvxx,T321=-vgyuy-vg(y)uyy,X331=-3g(y)uy2vx+3g(y)uxyuyv+3uxvgyuy+3uxvg(y)uyy-vgyuxxy-vg(y)uxxyy+g(y)uyvxxy-g(y)uxyvxy,Y331=g(y)vuty+g(y)vuxxxy-3g(y)vuxxuy-3g(y)vuxuxy-3g(y)uyuxvx+g(y)uyvt+g(y)uxxyvx,T331=-vgyuy-vg(y)uyy,X341=-3g(y)uy2vx+3g(y)uxyuyv+3uxvgyuy+3uxvg(y)uyy+g(y)uyvxxy+g(y)uxxyvy-g(y)uxyvxy,Y341=g(y)vuty-3g(y)vuxxuy-3g(y)vuxuxy-3g(y)uyuxvx+g(y)uyvt,T341=-vgyuy-vg(y)uyy,X411=-3g(y)uy2vx+3g(y)uxyuyv+3uxvgyuy+3uxvg(y)uyy-vgyuxxy-vg(y)uxxyy-vxxgyuy-vxxg(y)uyy+vxgyuxy+vxg(y)uxyy,Y411=g(y)vuxxxy-3g(y)vuxxuy-3g(y)vuxuxy-3g(y)uyuxvx+g(y)uyvxxx,T411=g(y)uyvy,X421=-3g(y)uy2vx+3g(y)uxyuyv+3uxvgyuy+3uxvg(y)uyy-vgyuxxy-vg(y)uxxyy+g(y)uyvxxy+vxgyuxy+vxg(y)uxyy,Y421=g(y)vuxxxy-3g(y)vuxxuy-3g(y)vuxuxy-3g(y)uyuxvx-g(y)uxyvxx,T421=g(y)uyvy,X431=-3g(y)uy2vx+3g(y)uxyuyv+3uxvgyuy+3uxvg(y)uyy-vgyuxxy-vg(y)uxxyy+g(y)uyvxxy-g(y)uxyvxy,Y431=g(y)vuxxxy-3g(y)vuxxuy-3g(y)vuxuxy-3g(y)uyuxvx+g(y)uxxyvx,T431=g(y)uyvy,X441=-3g(y)uy2vx+3g(y)uxyuyv+3uxvgyuy+3uxvg(y)uyy+g(y)uyvxxy+g(y)uxxyvy-g(y)uxyvxy,Y441=-3g(y)vuxxuy-3g(y)vuxuxy-3g(y)uyuxvx,T441=g(y)uyvy.
For the Lie symmetry V2, we can also get sixteen conservation laws by Theorem 4. For example, making use of
(23)(X21,Y21,T21)=(X2,Y2,T2)+(B1X,B1Y,0),
we can get
(24)X212=-3F(t)uyvx-3F(t)uxyv-3F(t)uxvy,Y212=3F(t)uxxv-Ftv+F(t)vxxx,T212=F(t)vy.
For the Lie symmetry V3, we can also get sixteen conservation laws by Theorem 4. For example, making use of
(25)(X42,Y42,T42)=(X4,Y4,T4)+(B2X,B2Y,0),
we can get
(26)X423=f(t)vuty-xftuyvx-3f(t)uyvxux+uyvft+13xftvxxy+f(t)uxvxxy+f(t)uxxyvx,Y423=-xftuxvx-3f(t)ux2vx-13xvftt-vftux-vf(t)utx-13ftvxx-f(t)vxxuxx,T423=13xftvy+f(t)vyux.
Using the Lie symmetry V4 and Theorem 4, sixteen conservation laws for (1) can be obtained. For example, making use of
(27)(X13,Y13,T13)=(X1,Y1,T1)+(B3X,B3Y,0),
we can get
(28)X134=-h(t)vuxxyt+13uhtvxxy-h(t)utxvxy-htvuxxy-xhtuxuyvx-19xhttvxy+h(t)vxxyut-23htuxvxy+118x2httvxxy-2xhtvuxuxy-htuuyvx-htuuxyv-htuuxvy-16x2httuyvx-16x2httuxyv-16x2httuxvy-xhtux2vy-3h(t)utuyvx-3h(t)utuxyv-3h(t)utuxvy+2htvuyux+13xvuyhtt+3h(t)vuyutx+13xhtvxxyux-13xhtvxyuxx+13xhtvuty,Y134=htvuuxx+16x2vhttuxx+2xhtvuxxux+3h(t)vuxxut+13uvtht+118x2vthtt+13xhtvtux+h(t)vtut+2htvux2+13xhttvux+3h(t)uxvutx+htuxxvx+19vxhtt+13xhtvxuxxx+h(t)vxutxx,T134=h(t)vuxxxy-3h(t)vuxxuy-3h(t)vuxuxy-13vuyht-13vxuxyht.
In the previous expressions of conservation laws, v is a solution of (14). If we can find an exact solution v of (14), explicit conservation laws of the ANNV equation (1) can be obtained by substituting it with the previous expressions. For example,
(29)v=m(y)+n(t)
is a solution of (14) with m(y) and n(t) being two arbitrary functions. By that, nontrivial conservation laws of (1) can be obtained.
Remark 5.
It is pointed out that the previous conservation laws are all nontrivial. The accuracy of them has been checked by Maple software.
Remark 6.
The conservation laws of (1) obtained in this paper are different from each other and are all different from those in [20].