五阶Korteweg-de Vries方程的孤立子解及其在表面波和内波中的应用
Soliton solutions to the fifth-order Korteweg - de Vries equation and their applications to surface and internal water waves
摘要：
研究了五阶Korteweg-de Vries方程的孤波解，该方程除了传统的二次非线性项和三阶色散项外，还含有三次非线性项和五阶线性色散项，以及两个非线性色散项.导出了该方程的精确孤立波解，并研究了其振幅、宽度和速度对控制方程参数的依赖关系。结果表明，导出的解可以代表嵌入或规则孤子依赖于方程参数。非线性色散项可以极大地影响孤立波的存在，它们的性质（规则的或嵌入的），轮廓，极性和相对于小扰动的稳定性。我们表明，特别是，在某些情况下，嵌入的孤子可以是稳定的，即使在与规则孤子的相互作用。所得结果适用于流体中的表面波和内波，也适用于其它介质（等离子体、固体波导、具有微结构的弹性介质等）中的波。
英文摘要：
We study solitary wave solutions of the fifth-order Korteweg - de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived and the dependence of its amplitude, width and speed on the parameters of the governing equation are studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The nonlinear dispersive terms can drastically influence the existence of solitary waves, their nature (regular or embedded), profile, polarity, and stability with respect to small perturbations. We show, in particular, that in some cases embedded solitons can be stable even with respect to interactions with regular solitons. The results obtained are applicable to surface and internal waves in fluids, as well as to waves in other media (plasma, solid waveguides, elastic media with microstructure, etc.).
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