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The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces. (English) Zbl 1161.35050

The author studies the nonlinear cubic Schrödinger equation: \[ i\partial_t u(t,x)+\Delta u(t,x)= \varepsilon|u|^2u(t,x),\;\varepsilon= \pm1,\;u(0,x)= u_0(x)\in H^\sigma(M). \] \((M,g)\): Riemannian surface. \(M\) has the Fermi coordinate \((s,r)\in S^1\times(- r_0,r_0)\) near a periodic geodesic \(\gamma\). The Laplace-Beltrami operator \[ \Delta= \Delta_g:= a^{-1}\partial_s(a^{-1}\partial_s)+ a^{-1}\partial_r(a\partial_r). \] Proposition 1. \(0<\sigma< 1/4\). Assume that \(M\) has a stable and non-degenerated periodic geodesics. Then the Cauchy problem is not uniformly well-posed. That is, there exist times \(t_n\to 0\) and \(u_{i,n}(t)\in H^\sigma(M)\), \(i= 1,2\): \(\| u_{i,n}(0)\|\leq C\), \[ \lim_{n\to\infty}\| u_{1,n}(0)- u_{2,n}(0)\|= 0, \]
\[ \limsup_{n\to\infty}\| u_{1,n}(t_n)- u_{2,n}(t_n)\|\geq C/2, \] by the norm \(\|\cdot\|\) in \(H^\sigma(M)\). In order to prove the Proposition 1, the author studies the equation \(-\Delta u=\lambda u-\varepsilon|u|^2u\), with \(u=\delta h^{-1/4} e^{is/h}f(s,r,h)\), \(\delta=\kappa h^\sigma\), \(\kappa> 0\), \(0\leq \sigma\leq 1/4\), with \[ u\sim\sum_{j\geq 0} h^{j/2}u_j\text{ and }\lambda\sim\sum_{j\geq 0} h^{-2+j/2}\lambda_j. \] Result. \(u_p(s, r)=\delta h^{-1/4}\chi(r)e^{is/h}(v_0+ h^{1/2}v_1+\cdots+ h^{p/2} v_p)\) \((s,h^{-1/2}r)\) and \[ \lambda_p= h^{-2}- 2h^{-1}(E_0+ h^{1/2}E_1+\cdots+ h^{p/2} E_p). \]
\[ \| u_p\|_{L^2(M)}\sim\delta,~-\Delta u_p= \lambda_p u_p- \varepsilon|u_p|u_p+ h^{(p-1)/2} g_p(h), \]
\[ \| h^{(p-1)/2} g_p(h)\|_{H^n}<\delta h^{(p-1-2n)/2}. \] Proposition 2. Denote by \(v= \exp(-i\lambda_pt)u_p\), \(p= 3\). Then \(\| v\|_{H^\sigma}\sim 1\) and \(\|(u- v)(t_h)\|_{H^\sigma}\to 0\) when \(h\to 0\), where \(t_h\sim h^{1/2-2\sigma}\log(1/h)\).
The author takes up \((v^1, v^2)\) and \((u^1(t), u^2(t))\) in Proposition 2 associated with \((\kappa,\kappa_h)\), \(\kappa_h\to\kappa\), and shows the instability in Proposition 1.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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