Solitons and cavitons (the latter are localized solutions with singularities) for the nonlocal Whitham equations are studied. The fourth order differential equation for traveling waves with a parameter in front of the fourth derivative is reduced to a reversible Hamiltonian system defined on a two-sheeted four-dimensional space. Solutions of the system which stay on one sheet represent smooth solutions of the equation but those which perform transitions through the branching plane represent solutions with jumps. They correspond to solutions with singularities of the fourth order differential equation – breaks of the first and third derivatives but continuous even derivatives. The Hamiltonian system can have two types of equilibria on different sheets, they can be saddle-centers or saddle-foci. Using analytic and numerical methods we found many types of homoclinic orbits to these equilibria both with a monotone asymptotics and oscillating ones. They correspond to solitons and cavitons of the initial equation. When we deal with homoclinic orbits to a saddle-center, the values of the second parameter (physical wave speed) are discrete but for the case of a saddle-focus they are continuous. The presence of multiplicity of such solutions displays the very complicated dynamics of the system.

研究了非局部Whitham方程的孤子和腔体（后者是具有奇异性的局部解）。参数在四阶导数前面的行波的四阶微分方程简化为在二维四维空间上定义的可逆哈密顿系统。停留在一张纸上的系统的解表示方程的平滑解，而执行通过分支平面的过渡的解表示具有跳跃的解。它们对应于具有四阶微分方程奇异性的解，即一阶和三阶导数的中断但连续的偶数导数。哈密​​顿量系统在不同的工作表上可以具有两种类型的平衡，它们可以是鞍中心或鞍焦点。使用解析和数值方法，我们发现了达到这些平衡的多种同宿轨道，包括单调渐近轨道和振荡轨道。它们对应于初始方程的孤子和空子。当我们处理同鞍轨道上的同斜轨道时，第二个参数（物理波速）的值是离散的，但对于鞍形焦点，它们是连续的。此类解决方案的多样性显示了系统非常复杂的动态。