弹性杆波导中几类非线性演化方程及其孤波解和冲击波解
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摘要
二十世纪六十年代，自然科学的许多科学分支几乎不约而同地出现了非线性问题的研究热潮，诸方面的研究汇成了非线性的洪流，孤子、湍流、混沌、分形及复杂系统等新的物理现象被揭示，大大扩展了人们的视野，并导致了自然科学认识论和发展观的一场大革命。非线性科学已成为近代科学发展的一个重要标志，它是自然科学各科学分支共同关心的真正的基础性研究。非线性科学涉及到自然界诸多复杂现象，具有广阔应用前景。特别是非线性动力学和非线性波动的研究对于解决物理学、化学、生物学和地球物理学中遇到的复杂现象和问题有着极其重要的意义。
非线性科学发展中一个重要成就就是孤立子理论的建立。在许多非线性物理领域，已经发现一大批非线性演化方程具有孤立子解。这些方程的共同特征是具有无穷个守恒律、可用散射反演法解析求解、存在B(?)cklund变换、完全可积分等。孤立子典型的特征是在其传播过程中伴随有能量集聚，且孤立子间相互作用时表现出犹如粒子弹性碰撞一样的行为。这些特性已在流体力学、等离子体、光纤通讯等技术领域获得广泛应用。
固体力学在线性波的研究方面曾取得过辉煌的成就，为推动物理学中波动理论的发展做出过巨大贡献。近年来固体结构中非线性波的研究已开始受到关注。本文在综述了其它非线性物理领域孤立子理论的研究基础上，以弹性细杆波导为对象，考虑了固体结构中常出现的非线性源及粘性耗散效应、几何弥散性质等，研究了固体中几类非线性波的传播问题，取得了以下一些主要结果：
1．利用Hamilton变分原理，导出了计及有限变形和横向剪切及横向惯
In 1960s the research upsurge of the nonlinear problems appeared simultaneously in many science branches in natural science. The research in many aspects formed nonlinear onrush. Many new physical phenomena such as soliton, turbulence, chaos, fractal and complex system etc. were discovered, which greatly expanded the people's view, and leaded to a great revolution in epistemology and development viewpoint in natural science. The nonlinear science has been an important symbol in the development of modern science. It is the actual fundamental study jointly concerned by various science branches in natural science. The nonlinear science involves lots of complex phenomena in nature, and has broad application prospect. Especially the researches in nonlinear dynamics and nonlinear wave-motion have extraordinary significance in solving problems and complicated phenomena encountered in physics, chemistry, biology and geophysics.
The establishment of soliton theory is an important achievement in the development of nonlinear science. In many nonlinear physics fields, it has been found that large quantities of nonlinear evolution equations had soliton solutions. The common characteristic of these equations are that they have infinite conservation rules, and can be solved by inverse scattering method, and have Backlund transform, and can be completely integrated et al. The typical properties of soliton are that its propagation accompanies energy centralization and the interacting among solitons shows behaviors similar to
非线性科学发展中一个重要成就就是孤立子理论的建立。在许多非线性物理领域，已经发现一大批非线性演化方程具有孤立子解。这些方程的共同特征是具有无穷个守恒律、可用散射反演法解析求解、存在B(?)cklund变换、完全可积分等。孤立子典型的特征是在其传播过程中伴随有能量集聚，且孤立子间相互作用时表现出犹如粒子弹性碰撞一样的行为。这些特性已在流体力学、等离子体、光纤通讯等技术领域获得广泛应用。
固体力学在线性波的研究方面曾取得过辉煌的成就，为推动物理学中波动理论的发展做出过巨大贡献。近年来固体结构中非线性波的研究已开始受到关注。本文在综述了其它非线性物理领域孤立子理论的研究基础上，以弹性细杆波导为对象，考虑了固体结构中常出现的非线性源及粘性耗散效应、几何弥散性质等，研究了固体中几类非线性波的传播问题，取得了以下一些主要结果：
1．利用Hamilton变分原理，导出了计及有限变形和横向剪切及横向惯
In 1960s the research upsurge of the nonlinear problems appeared simultaneously in many science branches in natural science. The research in many aspects formed nonlinear onrush. Many new physical phenomena such as soliton, turbulence, chaos, fractal and complex system etc. were discovered, which greatly expanded the people's view, and leaded to a great revolution in epistemology and development viewpoint in natural science. The nonlinear science has been an important symbol in the development of modern science. It is the actual fundamental study jointly concerned by various science branches in natural science. The nonlinear science involves lots of complex phenomena in nature, and has broad application prospect. Especially the researches in nonlinear dynamics and nonlinear wave-motion have extraordinary significance in solving problems and complicated phenomena encountered in physics, chemistry, biology and geophysics.
The establishment of soliton theory is an important achievement in the development of nonlinear science. In many nonlinear physics fields, it has been found that large quantities of nonlinear evolution equations had soliton solutions. The common characteristic of these equations are that they have infinite conservation rules, and can be solved by inverse scattering method, and have Backlund transform, and can be completely integrated et al. The typical properties of soliton are that its propagation accompanies energy centralization and the interacting among solitons shows behaviors similar to
引文
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