短脉冲方程的正则和奇异脉冲解和波前解及可能的等熵行为：相平面、多重无穷级数和变分方法
Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Short-Pulse Equation: Phase-Plane, Multi-Infinite Series and Variational Approaches
摘要：
在本文中，我们采用三个最近的分析方法来调查的行波解的所谓的短脉冲方程（SPE）的家庭的一些成员的可能类。最近，新的相平面分析的应用首次表明，在某些参数制度的存在性破碎扭结波解决方案。其次，光滑的行波是使用最近的技术，以获得收敛的多无穷级数解的同宿（异宿）轨道的行波方程的SPE方程，以及其广义版本与任意系数。这些分别对应于原始偏微分方程的脉冲（扭结或冲击）解。与大多数非加速收敛级数不同，它以相对较少的项获得高精度。最后，利用变分方法得到了SPE偏微分方程的正则孤立波解和嵌入孤立波解。用于获得嵌入孤子的技术结合了几个最近的一般变分技术，因此它本身是热门的。一个不寻常的特点，孤立波在这里得到的是，我们能够得到他们的分析形式（在假设的试验函数的ananterior）。因此，进行了直接的误差分析，显示所产生的孤立波的准确性。考虑到孤波解在波动力学和非线性偏微分方程信息传播中的重要性，以及关于广义SPE方程族的解所知不多的事实，所得到的结果是新的和及时的。
英文摘要：
In this paper we employ three recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a family of so-called short-pulse equations (SPE). A recent, novel application of phase-plane analysis is first employed to show the existence of breaking kink wave solutions in certain parameter regimes. Secondly, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic (heteroclinic) orbits of the traveling-wave equations for the SPE equation, as well as for its generalized version with arbitrary coefficients. These correspond to pulse (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the SPE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized SPE equations considered here, the results obtained are both new and timely.
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