论文标题
半线性卷曲 - 折返波动方程的呼吸和流氓波
Breathers and rogue waves for semilinear curl-curl wave equations
论文作者
论文摘要
我们考虑了半线性卷曲 - 折曲波方程$ s（x）\ partial_t^2 U + \ nabla \ nabla \ nabla \ nabla \ nabla \ times u + q（x）u \ pm v（x） \ Mathbb {r}^3 \ Times \ Mathbb {r} $和任意$ P> 1 $。根据系数$ s，q，v $，我们可以证明存在三种类型的本地化解决方案：空间无限的时间周期性解决方案衰减至$ 0 $，在空间无限的解决方案，时间周期性的解决方案趋向于空间无穷大的非平凡概况，这两种类型（这两种类型都称为呼吸器），以及$ 0 $ 0 $ 0 $ 0 $ 0 $ 0.我们的解决方案是薄弱的解决方案，并采用梯度领域的形式。因此，它们属于夹层操作员的内核，因此由于系数上的结构假设，半线性波方程将减小到ode。由于ODE中的空间依赖性只是一个参数依赖性，因此我们可以通过相位平面技术分析ODE，从而确定上述局部波的存在。我们分析的值得注意的副作用是紧凑的支撑呼吸器的存在以及一个局部波解$ u（x，t）$已经生成了相移的解决方案$ u（x，x，t+b（x））的完整连续性，其中连续函数$ b：\ mathbb {r}^r}^3 \ to \ to \ to \ mathbb compribiiss pocribiss a d primise
We consider localized solutions of variants of the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0$ for $(x,t)\in \mathbb{R}^3\times\mathbb{R}$ and arbitrary $p>1$. Depending on the coefficients $s, q, V$ we can prove the existence of three types of localized solutions: time-periodic solutions decaying to $0$ at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to $0$ both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curl-operator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a parametric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution $U(x,t)$ already generates a full continuum of phase-shifted solutions $U(x,t+b(x))$ where the continuous function $b:\mathbb{R}^3\to\mathbb{R}$ belongs to a suitable admissible family.
