Q7
Second order differential equations
Qualitative and asymptotic analysis of solutions
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Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
For $(\ell, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}$, we set: $$f_{\alpha}(x) = f(x)\exp\left(i\frac{x^{2}}{4}\right), \quad \Psi_{\alpha}(t, x) = \frac{1}{\sqrt{t}} f_{\alpha}\left(\frac{x}{\sqrt{t}}\right)$$
(a) Does there exist $t > 0$ such that $\Psi_{\alpha}(t, .)$ is periodic?
(b) Express $f_{\alpha}^{\prime}, f_{\alpha}^{\prime\prime}$ and $|f_{\alpha}|$ in terms of $f, f^{\prime}, f^{\prime\prime}$ and $|f|$.
(c) Justify that for all $(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}$, we have $\Psi_{\alpha}(., x) \in C^{1}(\mathbb{R}^{+*}, \mathbb{C})$ and $\Psi_{\alpha}(t, .) \in C^{2}(\mathbb{R}, \mathbb{C})$, then prove that $\Psi_{\alpha}$ satisfies equation $(F_{\alpha})$: $$\left(F_{\alpha}\right) \quad i\frac{\partial \Psi_{\alpha}}{\partial t}(t, x) + \frac{\partial^{2} \Psi_{\alpha}}{\partial x^{2}}(t, x) + \frac{1}{2}\Psi_{\alpha}(t, x)\left(\alpha\left|\Psi_{\alpha}(t, x)\right|^{2} + \frac{1}{t}\right) = 0$$