The validity of the derivative NLS approximation for systems with cubic nonlinearities (Q6616059)

The authors consider the cubic Klein-Gordon equation\N\[\N \partial_t^2 u = \partial_x^2 u - u +\rho(\partial_x)u^3,\quad t,x\in \mathbb R,\tag{NLKG}\N\]\Nwhere \(\rho\) is the Fourier multiplier given by\N\[\N\rho(ik) = \frac{k^2-1}{k^2+1}.\N\]\NThe goal is to justify the approximation\N\[\Nu(x,t)\approx \varepsilon^{1/2} A(\varepsilon (x-c_g t),\varepsilon^2t)e^{i(k_0x-\omega_0t)} + c.c.,\N\]\Nwhere the envelope \(A\) solves the derivative nonlinear Schrödinger equation\N\[\N -2i\omega_0\partial_T A = (1-c_g^2)\partial_X^2 A -3i\partial_X(A|A|^2).\tag{DNLS}\N\]\NCompared to the previous contribution by the authors [Z. Angew. Math. Phys. 74, No. 6, Paper No. 224, 20 p. (2023; Zbl 1533.35313)], the main feature is that the approximation is justified for Sobolev regularity, as opposed to Gevrey regularity. The main result states that for \(s\ge 12\), if \(A\in C([0,T],H^s)\) is a solution to (DNLS), then there exist \(\varepsilon_0>0\) and \(C>0\) such that if \(0<\varepsilon<\varepsilon_0\), (NLKG) has a solution\N\[\Nu,\partial_t u \in C([0,T/\varepsilon^2],H^1)\cap C^1([0,T/\varepsilon^2],L^2)\N\]\Nsuch that\N\[\N\sup_{0\le t\le T/\varepsilon^2} \sup_{x\in \mathbb R}\left | u(x,t) - \left( \varepsilon^{1/2} A(\varepsilon (x-c_g t),\varepsilon^2t)e^{i(k_0x-\omega_0t)} + c.c.\right)\right|\le C\varepsilon^{3/2}.\N\]\NA central difficulty of the paper is the analysis of resonances and their connection with the normal form approach.\N\NThe results presented here would certainly deserve some comparison with those gathered in e.g. [\textit{D. Lannes}, Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 2, 253--286 (2011; Zbl 1219.35283)] or [\textit{A. D. Bandrauk} (ed.) et al., Laser filamentation. Mathematical methods and models. Cham: Springer. 19--75 (2016; Zbl 1336.78001)], and references therein which, oddly enough, are not mentioned.