论文标题
长时间的渐近远渐进式渐近线方程，
Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-time
论文作者
论文摘要
我们研究了聚焦fokas-lenells（FL）方程的长期渐近行为$$ U_ {XT}+αβ^2U-2IαβU_X-x-αu_{XX}-Iαβ^2 | U | U | U | U |^2U_X = 0 \ label {CS} $ prient s sobolev sobolev sobolev sobolev primit solit soliT solits soliT solit。 FL方程是众所周知的Schrodinger方程的一般概括，也与衍生物Schrodinger模型相关联，但它表现出与它们的几种不同特征。 （i）FL方程的松弛对涉及$ k = 0 $的额外光谱奇点。 （ii）在渐近分析期间将出现四个固定相点，这需要更详细的必要描述才能获得聚焦FL方程的长期渐近学。基于Riemann-hilbert的问题针对焦点FL方程的初始值问题，我们表明，在任何固定的时空间锥$$ \ Mathcal {C} \ left（x_ {1}，x_ {2}，x_ {2}，v_ {1}，v_ {1}，v_ {1}，v_ {2} \ right）= \ ews few weft \ in \ Mathbb {r}^{2} | x = x_ {0}+v t，x_ {0} \ in \ left [x_ {1}，x_ {2} \ right]，v \ in \ left [v_ {1}，v_ {1}，v_ {2} \ right] \ right] \ right \}，$ pote the as usy asmpt a and coptiation $ a and coptiation $ a in coption $ use $ u（ $ n（\ MATHCAL {i}）$ - soliton在离散频谱上和领先的订单项$ \ Mathcal {o}（| t |^{ - 1/2}）$连续频谱上，最多可残留错误订单$ \ Mathcal $ \ Mathcal {O}（O}（O}）主要工具是$ \叠加{\ partial} $非线性陡峭下降方法和$ \叠加{\ partial} $ - 分析。
We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation $$ u_{xt}+αβ^2u-2iαβu_x-αu_{xx}-iαβ^2|u|^2u_x=0 \label{cs} $$ with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at $k=0$. (ii) four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone $$\mathcal{C}\left(x_{1}, x_{2}, v_{1}, v_{2}\right)=\left\{(x, t) \in \mathbb{R}^{2} | x=x_{0}+v t, x_{0} \in\left[x_{1}, x_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\},$$ the long-time asymptotic behavior of the solution $u(x,t)$ for the focusing FL equation can be characterized with an $N(\mathcal{I})$-soliton on discrete spectrums and a leading order term $\mathcal{O}(|t|^{-1/2})$ on continuous spectrum up to a residual error order $\mathcal{O}(|t|^{-3/4})$. The main tool is the $\overline{\partial}$ nonlinear steepest descent method and the $\overline{\partial}$-analysis.
