Conservation Laws of Two (2 + 1)-Dimensional Nonlinear Evolution Equations with Higher-Order Mixed Derivatives

and Applied Analysis 3 FromTheorem 3, we get the general formula of conservation laws for the system consisting of (1) and (14): X = ξL +W( ∂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 ∂uy − Dx ( ∂L ∂uxy ) − Dt ( ∂L ∂uyt ) −Dxxx ( ∂L ∂uxxxy )) + Dx (W)( ∂L ∂uxy + Dxx ( ∂L ∂uxxxy ))


Introduction
The construction of explicit forms of conservation laws plays an important role in the study of nonlinear science, as they are used for the development of appropriate numerical methods and for mathematical analysis, in particular, existence, uniqueness, and stability analysis [1][2][3].In addition, the existence of a large number of conservation laws of a partial differential equation (system) is a strong indication of its integrability.The famous Noether's theorem [4] has provided a systematic way of determining conservation laws for Euler-Lagrange equations, once their Noether symmetries are known, but this theorem relies on the availability of classical Lagrangians.To find conservation laws of differential equations without classical Lagrangians, researchers have made various generalizations of Noether's theorem [5][6][7][8][9][10][11][12][13][14][15][16].Among those, the new conservation theorem given by Ibragimov [5] is one of the most frequently used methods.
For any linear or nonlinear differential equations, Ibragimov's new conservation theorem offers a procedure for constructing explicit conservation laws associated with the known Lie, Lie-Backlund, or nonlocal symmetries.Furthermore, it does not require the existence of classical Lagrangians.Using the conservation laws formulas given by the theorem, conservation laws for lots of equations have been studied [6][7][8][9][10][11][12][13][14][15][16].When applying Ibragimov's theorem to a given nonlinear evolution equation with mixed derivatives, we must be careful with the mixed derivatives.If we apply the conservation laws formulas to equations with mixed derivatives directly, it will result in errors.In [9], we have proposed two modification rules to apply Ibragimov's theorem to study conservation laws of two evolution equations with mixed derivatives, but the mixed derivatives are all second order and not the highest derivative term.In this paper, we will give two new modification rules and then use Ibragimov's theorem to study conservation laws of the following ANNV equation [17][18][19][20]: and KP-BBM equation [21][22][23][24]   +   − 2 2  − 2  −   +   = 0, (2) where , , and  are constants.Both in (1) and ( 2), the highest derivative terms   and   are mixed.Furthermore, there are other lower-order mixed derivatives in addition to the higher-order mixed derivatives.The rest of the paper is organized as follows.In Section 2, we recall the main concepts and theorems used in this paper.In Section 3, taking the ANNV equation as an example, we first give two new modification rules which ensure the theorem can be applied to nonlinear evolution equations

Preliminaries
In this section, we briefly present the main notations and theorems [5][6][7] used in this paper.Consider an th-order nonlinear evolution equation with  independent variables  = ( is the total differential operator with respect to   . Definition 1.The adjoint equation of ( 3) is defined by with where denotes the Euler-Lagrange operator, V is a new dependent variable, and V = V().
In the following we recall the "new conservation theorem" given by Ibragimov in [5].
The conserved vector is given by where  is determined by (9),  is the Lie characteristic function, and

Two Modification Rules and Conservation Laws for the ANNV Equation
The asymmetric Nizhnik-Novikov-Veselov (ANNV) equation ( 1) is equivalent to the ANNV system [17,18] by the transformation  =   ,  =   .A series of new double periodic solutions to the system (13) were derived in [17], and the variable separation solutions of (13) have been given in [18].The Lie symmetry, reductions, and new exact solutions of the ANNV equation ( 1) have been studied by us from the point of Lax pair [19].Optimal system of groupinvariant solutions and conservation laws of (1) have been studied by Wang et al. [20].In the following, we will study the conservation laws of (1) by Theorem 3.

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 where V is a new dependent variable with respect to , , and .
According to Theorem 2, the formal Lagrangian for the system consisting of (1) and ( 14) is where V is a solution of (14).Suppose that the Lie symmetry for the ANNV equation ( 1) is as follows: From Theorem 3, we get the general formula of conservation laws for the system consisting of ( 1) and ( 14): )) where  is the Lie characteristic function,  =  −   −   −   , and  is the formal Lagrangian determined by (15).In fact, because of the existence of the mixed derivative terms   ,   , and   , the general formula of conservation laws must be modified; otherwise the previous , , and  do not satisfy The rules of modifications are as follows.
(1) In one conservation vector (, , or ), 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  or in , /  can only appear once; /  can only appear once in  and can appear three times in ; /  can only appear once in  and can appear three times in ; /  can only appear once in  and can appear two times in .
(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 (−  (/  )) in , then the term appears in  can only be   ()(/  ) and the term (−  (/  )) cannot appear in  at the same time.And if there is (−  (/  )) in , then the terms that appear in  contain   ()(/  ) and first and second total derivatives of /  .
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 /  , /  , and /  , the symmetry provides sixteen different conservation laws for the system consisting of (1) and (14).The conserved vectors are given as follows: with )) , where  is the Lie characteristic function and  =  −   −   −   ,  is the formal Lagrangian determined by (15).

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: where (), (), (), and ℎ() are arbitrary functions.
Using the Lie symmetry  1 and Theorem 4, we can get sixteen conservation laws for the system consisting of (1) and (14).They are listed as follows: For the Lie symmetry  2 , we can also get sixteen conservation laws by Theorem 4. For example, making use of we can get For the Lie symmetry  3 , we can also get sixteen conservation laws by Theorem 4. For example, making use of we can get Using the Lie symmetry  4 and Theorem 4, sixteen conservation laws for (1) can be obtained.For example, making use of we can get 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 Abstract and Applied Analysis 7 obtained by substituting it with the previous expressions.For example, is a solution of ( 14) with () and () 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].

Formulas of Conservation Laws and Explicit Conservation Laws for the KP-BBM Equation
The solutions of the KP-BBM equation ( 2) have been studied by Wazwaz in [21,22] who used the sine-cosine method, the tanh method, and the extended tanh method.Abdou [23] used the extended mapping method with symbolic computation to obtain some periodic solutions, solitary wave solution, and triangular wave solution.Exact solutions and conservation laws of (2) have been studied by Adem and Khalique using the Lie group analysis and the simplest equation method [24].

Formulas of Conservation Laws of the KP-BBM Equation.
To search for conservation laws of (2) by Theorem 3, Lie symmetry, formal Lagrangian, and adjoint equation of (2) must be known.According to Definition 1, the adjoint equation of ( 2) is where V is a new dependent variable with respect to , , and .
According to Theorem 2, the formal Lagrangian for the system consisting of ( 2) and ( 30) is Since there are a higher-order mixed derivative   and a mixed derivative   in (2), the two modification rules must be used if we want to get conservation laws of (2) by Theorem 3. Therefore, we can get the following statement.

Theorem 7. Suppose that the Lie symmetry of the KP-BBM equation is as follows:
According to the different locations of /  and /  , the symmetry provides eight different conservation laws for the system consisting of (2) and (30).The conserved vectors are given as follows: (  ,   ,   ) = (  ,  )) , where  is the Lie characteristic function,  = −  −  −   , and  is the formal Lagrangian determined by (31).

Explicit Conservation Laws of the KP-BBM Equation.
Lie symmetries of ( 2) have been derived in [24] and are listed as follows: Applying Theorem 7, we can obtain conservation laws for the system consisting of ( 2) and (30).For the symmetry  1 , we can get the following eight conservation laws: where For the symmetry  2 , we can also get eight conservation laws for the system of ( 2) and (30).For example, making use of we can get Similarly, for the symmetry  3 , we can get eight conservation laws for the system of ( 2) and (30).For example, making use of we can get For the symmetry  4 , we only list the conservation laws derived by and they are as follows: In the previous expressions of conservation laws, V is a solution of the adjoint equation (30).If we can find an exact solution V of (30), explicit conservation laws for the KP-BBM equation ( 2) can be obtained by substituting it with the previous expressions.For example, V = ( +  ())  +  () is a solution of (30) with () and () being two arbitrary functions.By that we can get many infinite conservation laws for (2).Furthermore, the conservation laws are nontrivial and time dependent.
Remark 8.The correctness of the conservation laws of (2) obtained here has been checked by Maple software.The conservation laws obtained here for (2) are much more than those in [24] and different from them.

Concluding Remarks
Recently, conservation laws of nonlinear evolution equations with mixed derivatives have attracted the interest of mathematical and physical researchers.As shown in [16], when applying Noether's theorem and partial Noether's theorem to obtain conservation laws of nonlinear evolution equations with higher-order mixed derivatives, the obtained conservation laws must be adjusted to satisfy the definition of conservation laws.We face the same problem when applying Ibragimov's new conservation theorem to find conservation laws of nonlinear evolution equations with mixed derivatives.
In this paper, we propose two modification rules which ensure that Ibragimov's theorem can be applied to nonlinear evolution equations with higher-order and lower-order mixed derivatives.The two modification rules given in this paper are a generalization of those proposed in [9].The results are used to study the conservation laws of two partial differential equations with higher-order mixed derivatives: the ANNV equation and the KP-BBM equation.Many infinite explicit and nontrivial conservation laws are obtained for the two equations.Based on the two modification rules, Ibragimov's new conservation theorem can be used to find conservation laws of other partial differential equations with any mixed derivatives.