^{1}

In this paper, conservation laws for the

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 [

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 [

The rest of the paper is organized as follows. In Section

In this section, we briefly present the main notations and theorems [

The adjoint equation of (

The system consisting of (

In the following we recall the “new conservation theorem” given by Ibragimov in [

Any Lie point, Lie-Backlund, and nonlocal symmetries,

The asymmetric Nizhnik-Novikov-Veselov (ANNV) equation (

To search for conservation laws of (

According to Theorem

Suppose that the Lie symmetry for the ANNV equation (

In fact, because of the existence of the mixed derivative terms

Applying the two rules to the general conservation laws formula in Theorem

Suppose that the Lie symmetry of the ANNV equation (

Now, conservation laws of (

Using the Lie symmetry

For the Lie symmetry

For the Lie symmetry

Using the Lie symmetry

In the previous expressions of conservation laws,

It is pointed out that the previous conservation laws are all nontrivial. The accuracy of them has been checked by Maple software.

The conservation laws of (

The solutions of the KP-BBM equation (

To search for conservation laws of (

According to Theorem

Since there are a higher-order mixed derivative

Suppose that the Lie symmetry of the KP-BBM equation is as follows:

Lie symmetries of (

For the symmetry

Similarly, for the symmetry

For the symmetry

In the previous expressions of conservation laws,

The correctness of the conservation laws of (

Recently, conservation laws of nonlinear evolution equations with mixed derivatives have attracted the interest of mathematical and physical researchers. As shown in [

This study is supported by the National Natural Science Foundation of China (Grant no. 10871117) and the Natural Science Foundation of Shandong Province (Grant no. ZR2010AL019).