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本文讨论地球流体运动浅水波模式中非频散的周期解以及孤立波产生的可能性。首先,利用常微分方程的定性理论得到了浅水波存在的条件与浅水波解的解析表达式。在问题退化的过程中,发现当某物理量产生非线性孤立波解时,必伴随着其它物理量无界,因此认为系统根本不存在所谓的孤立波解。接着本文引进了广义能量(即拟能)的概念,指出当外界特定影响促使拟能产生微小变化时,此时系统会产生孤立波,最后得到了区别KdV方程的孤立波解的解析表达式。
This paper discusses the periodic solutions of the non-dispersion and the possibility of solitary waves in the shallow-water wave mode of the Earth’s fluid motion. First of all, by using the qualitative theory of ordinary differential equations, the conditions for the existence of shallow water waves and the analytical expressions of shallow water wave solutions are obtained. In the process of problem degeneration, we find that when a nonlinear solitary wave solution is generated for a given physical quantity, it must be accompanied by unboundedness of other physical quantities. Therefore, there is no so-called solitary wave solution in the system at all. Then, the concept of generalized energy (quasi-energy) is introduced in this paper. It is pointed out that when a particular influence of the outside world is caused by a small change, the system will generate solitary waves. Finally, an analytical expression of solitary wave solutions to KdV equation is obtained.