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.

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。