Qin Mao-Chang

Non-Noether conserved quantity constructed by using form invariance for Birkhoffian system

Based on the invariance of Birkhoffian equations under the infinitesimal transformations of groups, the definition and the criterion of a form invariance for a Birkhoffian system are established. The condition under which the form invariance can lead to a non-Noether conserved quantity and the form of the conserved quantity are deduced by relying on the total time derivative along the trajectory of the equations and two corollaries in special cases are presented. An example is finally given to illustrate the application of the results.

Unified Symmetry of Hamilton Systems

The definition and the criterion of a unified symmetry for a Hamilton system are presented. The sufficient condition under which the Noether symmetry is a unified symmetry for the system is given. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, is obtained. An example is finally given to illustrate the application of the results.

Unified symmetry of holonomic mechanical systems

The definition and the criterion of a unified symmetry for a holonomic mechanical system are presented based on the total time derivative along the trajectory of the system. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, is obtained. An example is given to illustrate the application of the results.

An Effective Method for Seeking Conservation Laws of Partial Differential Equations

This paper introduces an effective method for seeking local conservation laws of general partial differential equations (PDEs). The well-known variational principle does not involve in this method. Alternatively, the conservation laws can be derived from symmetries, which include the symmetries of the associated linearized equation of the PDEs, and the adjoint symmetries of the adjoint equation of the PDEs.

Nonclassical Potential Symmetries and New Explicit Solutions of Burgers Equation

A new method is used to determine the nonclassical potential symmetry generators of Burgers equation. Some classes of new explicit solutions, which cannot be obtained by Lie symmetry group of Burgers equation or its integrated equation, are obtained by using these new nonclassical potential symmetry generators.

Conserved Densities of the Black-Scholes Equation

A class of new conserved densities of the Black-Scholes equation are constructed by using the multiplier that is derived from the result of divergence expression annihilation under the full Euler operator. The method does not depend on the symmetries of the Black-Scholes equation. These conserved densities can be expressed by solutions of the classical heat equation.

New Symmetries for a Model of Fast Diffusion

The new symmetries for a mathematical model of fast diffusion are determined. A new system method is given to search for new symmetries of differential equations written in a conserved form, several new symmetry generators and exact solutions are presented.