Mathematical Sciences: Mathematical Physics and Continuum Mechanics

这项研究项目致力于连续介质力学和数学物理方程的各种代数和几何方面的研究。主要的集中领域包括线性和非线性弹性方程、李代数和准精确可解的量子问题、Cartan等价方法、双哈密顿系统、孤立波和非线性耗散。具体应用包括：1.线性和非线性弹性力学变分问题的守恒律、对称性、标准形和不变量；2.微分算子和拟精确可解薛定谔算子的李代数的分类；3.双哈密顿系统的标准形和可积性；4.常微分方程和偏微分方程解的非线性耗散和规范形；5.变分中的等价问题；6.高阶模型发展方程孤立波解的存在性和不存在性。这些研究课题与流体力学、弹性力学和量子力学产生的重要问题的研究有关。要解决的特殊问题包括研究某些水波模型方程的孤立波解，一类新的和物理上重要的量子力学系统的分类和应用，了解非线性系统数学模型中的耗散(摩擦和粘性)，以及线性和非线性弹性的对称性和守恒定律的分析。

This research project is devoted to various algebraic and geometric aspects of the equations of continuum mechanics and mathematical physics. The principal areas of concentration include the equations of linear and nonlinear elasticity, Lie algebras and quasi-exactly solvable quantum problems, the Cartan equivalence method, bi-Hamiltonian systems, solitary waves, and nonlinear dissipation. Specific applications include: 1. Conservation laws, symmetries, canonical forms and invariants for the variational problems of linear and nonlinear elasticity, 2. The classification of Lie algebras of differential operators and quasi-exactly solvable Schrodinger operators, 3. Canonical forms and integrability of bi-Hamiltonian systems, 4. Nonlinear dissipation and normal forms for both ordinary and partial differential equations, 5. Equivalence problems from the calculus of variations, and 6. Existence and non-existence of solitary wave solutions to higher order model evolution equations. These research topics are relevant to the studies of important problems arising from fluid mechanics, elasticity, and quantum mechanics. Particular problems to be addressed include the investigation of solitary wave solutions to certain model equations for water waves, classification and application of a new and physically important class of quantum mechanical systems, understanding dissipation (friction and viscosity) in mathematical models of nonlinear systems, and the analysis of symmetry and conservation laws for linear and nonlinear elasticity.