Communications in Mathematical Physics ( IF 2.2 ) Pub Date : 2021-02-26 , DOI: 10.1007/s00220-021-03946-x Anne Boutet de Monvel , Jonatan Lenells , Dmitry Shepelsky
We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching two different plane waves \(A_j\mathrm {e}^{\mathrm {i}\phi _j}\mathrm {e}^{-2\mathrm {i}B_jx}\), \(j=1,2\) as \(x\rightarrow \pm \infty \). Using Riemann–Hilbert techniques and Deift–Zhou steepest descent arguments, we study the long-time asymptotics of the solution. We detect that each of the cases \(B_1<B_2\), \(B_1>B_2\), and \(B_1=B_2\) deserves a separate analysis. Focusing mainly on the first case, the so-called shock case, we show that there is a wide range of possible asymptotic scenarios. We also propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors.
中文翻译:
具有阶跃振荡背景的聚焦NLS方程:长时间渐近情形
对于初始数据接近两个不同平面波\(A_j \ mathrm {e} ^ {\ mathrm {i} \ phi _j} \ mathrm {e} ^ {-2 \ mathrm { i} B_jx} \),\(j = 1,2 \)为\(x \ rightarrow \ pm \ infty \)。使用黎曼–希尔伯特技术和戴夫特–周最陡下降参数,我们研究了该解的长期渐近性。我们检测到每种情况\(B_1 <B_2 \),\(B_1> B_2 \)和\(B_1 = B_2 \)值得单独分析。我们主要关注第一种情况,即所谓的冲击情况,我们表明存在多种可能的渐近情况。我们还提出了一种方法,用于严格确定某些更高类的渐近扇形的存在性。