arXiv 0705.31592007-05-21nlin.SInlin.PS

sine-Gordon方程半经典非特征Cauchy问题的精确解

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

Robert Buckingham Peter D. Miller

备注：: 49 pages, 10 figures

摘要：

利用sine-Gordon方程作为约瑟夫森结中磁通量传播的模型，研究了在色散参数$\e$趋于零的半经典极限下该方程的初值问题。假设自然的初始数据具有的轮廓的移动$-2π$扭结在时间为零，我们分析计算的散射数据，这个完全可积柯西问题的所有$\e>0$足够小，并进一步反演散射变换计算的解决方案，为一个序列的任意小的$\e$。这个精确解序列类似于聚焦非线性薛定谔方程的著名的$N$ -孤子（或高阶孤子）解。小$\e$的精确解图揭示了半经典极限中出现的某些特征。例如，在极限$ε\to 0$中，观察到非线性焦散的出现。在附录中，我们从逆散射和经典分析（Picard迭代）的角度给出了柯西问题的自包含说明。具体而言，附录A包含通用$L^1$ -Sobolev初始数据的反散射方法的完整公式，附录B建立了$L^p$ -Sobolev初始数据的适定性（特别是完全证明了附录A中的反散射分析）。

英文摘要：

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter $\e$ tends to zero. Assuming natural initial data having the profile of a moving $-2π$ kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all $\e>0$ sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small $\e$. This sequence of exact solutions is analogous to that of the well-known $N$-soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. Plots of exact solutions for small $\e$ reveal certain features that emerge in the semiclassical limit. For example, in the limit $ε\to 0$ one observes the appearance of nonlinear caustics. In the appendices we give a self contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic $L^1$-Sobolev initial data, and Appendix B establishes the well-posedness for $L^p$-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).

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