# Comment on “Conservation Laws of Two  (2 + 1)-Dimensional Nonlinear Evolution Equations with Higher-Order Mixed Derivatives”

Authors: Long Wei; Yang Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

In a recent paper (Zhang (2013)), the author claims that he has proposed two rules to modify Ibragimov’s theorem on conservation laws to “ensure the theorem can be applied to nonlinear evolution equations with any mixed derivatives.”  In this letter, we analysis the paper.  Indeed, the so-called “modification rules” are needless and the theorem of Ibragimov can be applied to construct conservation laws directly for nonlinear equations with any mixed derivatives as long as the formal Lagrangian is rewritten in symmetric form. Moreover, the conservation laws obtained by the so-called “modification rules” in the paper under discussion are equivalent to the one obtained by Ibragimov’s theorem.


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## Body

## 1. Introduction

Noether’s famous theorem about symmetries and conservation laws is now almost a century old and has been discussed in literally hundreds of papers and is covered in many textbooks, for example, in [1, 2]. Noether’s theorem allows one to construct conservation laws for differential equations following a straightforward algorithm. Although Noether’s approach provides an elegant algorithm for finding conservation laws, it possesses a strong limitation: it can only be applied to equations having variational structure. However, a large number of differential equations without variational structure admit conservation laws. Recently, Ibragimov proved a result in [3] which allows one to construct conservation laws for equations without variational structure. Essentially, Ibragimov’s theorem is an extension of Noether’s theorem by introducing formal Lagrangian to get rid of the variational limitation. There have been abundant papers on constructing conservation laws using Ibragimov’s theorem; see [4–9] and references therein.In [10], to construct conservation laws for some nonlinear equations with higher-order mixed derivatives, the author proposed two so-called “modification rules” to modify Ibragimov’s theorem and gave “new” formulae of conservation laws for ANNV equation and KP-BBM equation. The two rules in [10] are uncalled for and illusive; indeed, Ibragimov’s theorem can be applied to construct conservation laws directly for nonlinear equations with higher-order mixed derivatives. To do so, we only need to rewrite the corresponding form Lagrangian in symmetric form. Moreover, by direct calculations, we find that the conservation laws obtained by so-called “modification rules” in [10] are equivalent to the one obtained by Ibragimov’s theorem.For simplicity, we only illuminate our statement for the ANNV equation [10, (1)] in the following section.


## 2. Formulae of Conservation Laws for ANNV Equation and the Equivalence

In this section, we first apply Ibragimov’s theorem to construct conservation laws for ANNV equation [10, (1)]
(1)uyt+uxxxy-3uxxuy-3uxuxy=0.
The form Lagrangian of (1) and the corresponding adjoint equation is
(2)L=v(uyt+uxxxy-3uxxuy-3uxuxy),
where v is the solution of the adjoint equation. We rewrite L in the symmetric form as follows:
(3)ℒ=v[12(uyt+uty)ccccc+14(uxxxy+uxxyx+uxyxx+uyxxx)ccccc-3uxxuy-32ux(uxy+uyx)].
For any Lie point, Lie-Bäcklund, and nonlocal symmetry of (1),
(4)V=ξ∂∂x+η∂∂y+τ∂∂t+ϕ∂∂u;
from Ibragimov’s theorem, we obtain the general formulae of conservation laws for the system consisting of (1) and the adjoint equation as follows:
(5)X=ξℒ+W[ℒux-Dx(ℒuxx)-Dy(ℒuxy)ccccccccccc-Dxxy(ℒuxxxy)-Dxyx(ℒuxxyx)ccccccccccc-Dyxx(ℒuxyxx)]+Dx(W)[ℒuxx+Dxy(ℒuxxxy)+Dyx(ℒuxxyx)]+Dy(W)[ℒuxy+Dxx(ℒuxyxx)]+Dxx(W)[-Dy(ℒuxxxy)]+Dxy(W)×[-Dx(ℒuxxyx)]+Dyx(W)[-Dx(ℒuxyxx)]+Dxxy(W)ℒuxxxy+Dxyx(W)ℒuxxyx+Dyxx(W)ℒuxyxx,(6)Y=ηℒ+W[ℒuy-Dx(ℒuyx)-Dt(ℒuyt)cccccccccccc-Dxxx(ℒuyxxx)]+Dx(W)[ℒuyx+Dxx(ℒuyxxx)]+Dt(W)ℒuyt+Dxx(W)[-Dx(ℒuyxxx)]+Dxxx(W)ℒuyxxx,(7)T=τℒ+W(-Dyℒuty)+Dy(W)ℒuty,
where W=ϕ-ξux-ηuy-τut is the Lie characteristic function, ℒ is the formal Lagrangian in the symmetric form given by (3), and ℒw=∂ℒ/∂w. Thus, the conservation laws for (1) can be derived from above formulae (5), (6), and (7) if Lie symmetries of (1) are known. For example, let us construct the conserved vector corresponding to the generator
(8)V1=g(y)∂∂y
as follows:
(9)X1=32g(y)uyuxyv-3g(y)uy2vx-32g(y)uxuyvy+34g(y)uyvxxy-12g(y)uxyvxy-14g′(y)uyvxx+32g′(y)uxuyv-14g(y)uyyvxx+32g(y)uxuyyv+14g(y)uxxyvy+12vxg′(y)uxy+12g(y)vxuxyy-34g′(y)uxxyv-34g(y)uxxyyv,(10)Y1=14g(y)(2utyv+3vuxxxy-6vuxxuy-6vuxuxyccccccccc-6uxuyvx+2uyvt+uyvxxxcccccccccc-uxyvxx+uxxyvx),(11)T1=12g(y)uyvy-12g′(y)uyv-12g(y)uyyv.We should point out here that the conservation laws obtained in [10] (X11j,Y11j,T11j), (X2j1,Y2j1,T2j1), (X3j1,Y3j1,T3j1), (X4j1,Y4j1,T4j1) (j=1,2,3,4) are equivalent to our result (X1,Y1,T1) given above if we transfer the terms Dy(⋯), Dx(⋯) from X1, Y1 to each other and transfer the terms Dt(⋯), Dy(⋯) from Y1, T1 to each other, respectively.To explain the fact, in general, we show the equivalence of the formulae of conservation laws (5), (6), (7), and the ones (Xij,Yij,Tij) in Theorem 3.1 in [10].Now let us consider the formulae of conserved vector(X,Y,T) of (1). Observe the facts that in (5)
(12)-WDy(ℒuxy)+Dy(W)ℒuxy=Dy(Wℒuxy)-2WDy(ℒuxy),WDxxy(ℒuxxxy)=Dy[WDxx(ℒuxxxy)]-Dy(W)Dxx(ℒuxxxy),Dx(W)Dxyℒuxxxy=Dy[Dx(W)Dx(ℒuxxxy)]-Dxy(W)Dx(ℒuxxxy),Dxx(W)Dy(ℒxxxy)=Dy[Dxx(W)ℒuxxxy]-Dxxy(W)ℒuxxxy,WDxxy(ℒuxxxy)=WDxyx(ℒuxxyx)=WDyxx(ℒuxyxx),Dx(W)Dxyℒuxxxy=Dx(W)Dyxℒuxxyx,
in (6)
(13)-WDt(ℒuyt)+Dt(W)ℒuyt=Dt[Wℒuyt]-2WDt(ℒuyt),WDx(ℒuyx)=Dx(Wℒuyx)-Dx(W)ℒuyx,Dx(W)Dxx(ℒuyxxx)=Dx[WDxx(ℒuyxxx)]-WDxxx(ℒuyxxx),Dxx(W)Dx(ℒuyxxx)=Dx[Dx(W)Dx(ℒuyxxx)]-Dx(W)Dxx(ℒuyxxx),DxxxWℒuyxxx=Dx[Dxx(W)ℒuyxxx]-Dxx(W)Dx(ℒuyxxx),
and in (7)
(14)-WDy(ℒuty)+Dy(W)ℒuty=-Dy[Wℒuty]+2Dy(W)ℒuty=Dy[Wℒuty]-2WDy(ℒuty).
Thus, we have that
(15)X=ξℒ+Wℒux-WDx(ℒuxx)-2WDy(ℒuxy)+Dx(W)ℒuxx-4Dxy(W)Dx(ℒuxxxy)+4Dxxy(W)ℒuxxxy+4Dy(W)Dxxℒuxxxy+Dy[Wℒuxy]-3Dy[WDxx(ℒuxxxy)]+2Dy[Dx(W)Dx(ℒuxxxy)]-Dy[Dxx(W)ℒuxxxy],Y=ηℒ+Wℒuy-2WDt(ℒuyt)+2Dx(W)ℒuxy-4WDxxx(ℒuxxxy)-Dx[Wℒuyx]+3Dx[WDxx(ℒuxxxy)]-2Dx[Dx(W)Dx(ℒuxxxy)]+Dx[Dxx(W)ℒuxxxy]+Dt[Wℒuyt],T=τℒ+2Dy(W)ℒuty-Dy[Wℒuty].
Therefore, we can deduce the conserved quality as follows:
(16)X=ξℒ+Wℒux-WDx(ℒuxx)-2WDy(ℒuxy)+Dx(W)ℒuxx-4Dxy(W)Dx(ℒuxxxy)+4Dxxy(W)ℒuxxxy+4Dy(W)Dxxℒuxxxy,Y=ηℒ+Wℒuy-2WDt(ℒuyt)+2Dx(W)ℒuxy-4WDxxx(ℒuxxxy),T=τℒ+2Dy(W)ℒuty.
Rewriting above result in L, we have
(17)X=ξL+WLux-WDx(Luxx)-WDy(Luxy)+Dx(W)Luxx-Dxy(W)Dx(Luxxxy)+Dxxy(W)Luxxxy+Dy(W)DxxLuxxxy,Y=ηL+WLuy-WDt(Luyt)+Dx(W)Luxy-WDxxx(Luxxxy),T=τℒ+Dy(W)Luyt;
this is the result (X11,Y11,T11)=(X1,Y1,T1)+(B1X,B1Y,0) in [10].Similarly, if we rewrite (5), (6), and (7) in the following way:
(18)X=ξℒ+Wℒux-WDx(ℒuxx)+Dx(W)ℒuxx-4WDxxy(ℒuxxxy)+2Dy(W)ℒuxy+4Dx(W)Dxy(ℒuxxxy)+4Dxxy(W)ℒuxxxy-Dy[Wℒuxy]+Dy[WDxx(ℒuxyxx)]-Dy[Dxx(W)ℒuxyxx]-2Dy[Dx(W)Dx(ℒuxyxx)],Y=ηℒ+Wℒuy-2WDx(ℒuyx)+2Dt(W)ℒuyt-4Dxx(W)Dx(ℒuyxxx)+Dx[Wℒuyx]-Dx[WDxx(ℒuyxxx)]+Dx[Dxx(W)ℒuyxxx]+2Dx[Dx(W)Dx(ℒuyxxx)]-Dt[Wℒuyt],T=τℒ-2WDy(ℒuty)+Dy[Wℒuty],
then the conservation law (X,Y,T) can be deduced in L as follows:
(19)X=ξL+WLux-WDx(Luxx)+Dx(W)Luxx-WDxxy(Luxxxy)+Dy(W)Luxy+Dx(W)Dxy(Luxxxy)+Dxxy(W)Luxxxy,Y=ηL+WLuy-WDx(Luxy)+Dt(W)Luyt-Dxx(W)Dx(Luxxxy),T=τL-WDy(Luyt);
this is the conservation law (X43,Y43,T43)=(X4,Y4,T4)+(B3X,B3Y,0) in [10].Adopting the same procedure as above, we see that all the conservation laws(Xij,Yij,Tij) obtained in Theorem 3.1 in [10] can be deduced from our result (X,Y,T) given by (5), (6), and (7); we have checked them and omit the details here.


## 3. Conclusions

In this letter, we have analyzed the paper [10]. We see that the two so-called “modification rules” in [10] are needless and Ibragimov’s theorem can be applied to construct conservation laws directly for nonlinear equations with higher-order mixed derivatives. By direct calculations, we have found that the conservation laws obtained by the so-called “modification rules” in [10] are equivalent to the one obtained by Ibragimov’s theorem. Therefore, Ibragimov’s theorem on conservation laws need not be modified.



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*Source: 264049-2014-01-28.xml*