Burde G I
The Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 84990, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Aug;84(2 Pt 2):026615. doi: 10.1103/PhysRevE.84.026615. Epub 2011 Aug 26.
Some effects in the soliton dynamics governed by higher-order Korteweg-de Vries (KdV) type equations are discussed. This is done based on the exact explicit solutions of the equations derived in the paper. It is shown that some higher order KdV equations possessing multisoliton solutions also admit steady state solutions in terms of algebraic functions describing localized patterns. Solutions including both those static patterns and propagating KdV-like solitons are combinations of algebraic and hyperbolic functions. It is shown that the localized structures behave like static solitons upon collisions with regular moving solitons, with their shape remaining unchanged after the collision and only the position shifted. These phenomena are not revealed in common multisoliton solutions derived using inverse scattering or Hirota's method. The solutions of the higher-order KdV type equations were obtained using a method devised for obtaining soliton solutions of nonlinear evolution equations. This method can be combined with Hirota's method with a modified representation of the solution which allows the results to be extended to multisoliton solutions. The prospects for applying the methods to soliton equations not of KdV type are discussed.
讨论了由高阶Korteweg-de Vries（KdV）型方程所支配的孤子动力学中的一些效应。这是基于本文中推导的方程的精确显式解来进行的。结果表明，一些具有多孤子解的高阶KdV方程也允许以描述局域模式的代数函数形式存在稳态解。包括那些静态模式和传播的类KdV孤子的解是代数函数和双曲函数的组合。结果表明，局域结构在与规则移动孤子碰撞时表现得像静态孤子，碰撞后其形状保持不变，只是位置发生了偏移。这些现象在使用逆散射或Hirota方法得到的普通多孤子解中并未显现。高阶KdV型方程的解是使用一种为获得非线性演化方程的孤子解而设计的方法得到的。该方法可以与Hirota方法相结合，并对解进行修改表示，从而使结果能够扩展到多孤子解。还讨论了将这些方法应用于非KdV型孤子方程的前景。
