I have been struggling with some linearization argument of the following paper: "M. Weinstein: Modulational stability of ground states of NLS". In order to give a bit of context to my question, let us consider the NLS equation $$ 2i\phi_t+\Delta\phi+\vert\phi\vert^{2\sigma}\phi=0, \quad 0<\sigma<\tfrac{2}{n-2}.$$ This equation has very interesting "localized" solutions of the form: $$\phi(t,x)=u(x)e^{it/2},$$ where $u(t,x)$ solves $\Delta u-u+\vert u\vert^{2\sigma}u=0$. Besides, the latter equation has an even more interesting real, positive and radial $H^1(\mathbb{R}^n)$ solution called "Ground state" and denoted by $R(x)$.

Now let me try to explain my question. Consider the perturbed Initial Valued Problem (IVP): $$2i\phi_t^\varepsilon+\Delta \phi^\varepsilon+\vert \phi^\varepsilon\vert^{2\sigma}\phi^\varepsilon=\varepsilon F(\vert \phi^\varepsilon\vert)\phi^\varepsilon, \quad \phi^\varepsilon(t=0,x)=R(x)+\varepsilon S(x)$$ We will seek solutions of the previous equation of the form $$\phi^\varepsilon(t,x)=(R(x)+\varepsilon w_1+\varepsilon^2 w_2+...)e^{it/2}.$$ According to Weinstein if you reeplace this function into the perturbed equation and linearize you will get the following IVP for the linearized perturbation $w$: $$ 2iw_t+\Delta w-w+(\sigma+1)R^{2\sigma}w+\sigma R^{2\sigma}\overline{w}=F(R)R, \quad w(0,x)=0.$$ Now my problem is: I do not really understand how to obtain this linearization, can someone explain a little bit how to do it? Or recommend some references to learn about it. I tried replacing $\phi^\varepsilon$ (truncated after $\varepsilon w$) and then I took the derivative with respect to $\varepsilon$ and evaluate at $\varepsilon=0$, but I cannot recover the equation claimed by Weinstein, so I think that I'm not undersitanding how to linearize.

Note2: The parameter $n$ denotes the dimension.

  • $\begingroup$ your approach seems to be good. Try to do it again, substitute $\phi^\epsilon(t,x) = (R(x)+\epsilon w)e^{it/2}$ and get rid of all $\epsilon^2$ term. Also, use that $R(x)$ solves stationary NLS equation. $\endgroup$ – Jane May 30 at 18:29

If we have a nonlinear operator acting on some space of (weakly) differentiable functions, linearizing it at a given "point" means simply verify that its Gateaux derivative exists at that "point". In the case under analysis, we have that the nonlinear operator has the following form: $$ \mathfrak{F}(t,x,\phi^\varepsilon)=2i\phi_t^\varepsilon+\Delta \phi^\varepsilon+\vert \phi^\varepsilon\vert^{2\sigma}\phi^\varepsilon-\varepsilon F(\vert \phi^\varepsilon\vert)\phi^\varepsilon $$ Define $w_1(t,x)\triangleq w(t,x)$: noting that $$ \begin{split} \left.\phi^\varepsilon(t,x)\right|_{\varepsilon=0}&=R(x)e^{it/2}\iff \left. \overline{\phi^\varepsilon}(t,x)\right|_{\varepsilon=0}=R(x)e^{-it/2},\\ \\ \left.\frac{\partial}{\partial \varepsilon} \phi^\varepsilon(t,x)\right|_{\varepsilon=0} &= w(t,x)e^{it/2}\iff \left.\frac{\partial}{\partial \varepsilon} \overline{\phi^\varepsilon}(t,x)\right|_{\varepsilon=0} = \overline{w}(t,x)e^{-it/2},\\ \\ \left.\frac{\partial}{\partial \varepsilon} |\phi^\varepsilon(t,x)|^{2\sigma}\right|_{\varepsilon=0} & = \left.\frac{\partial}{\partial \varepsilon} \big(\phi^\varepsilon(t,x)\overline{\phi^\varepsilon}(t,x)\big)^{\sigma}\right|_{\varepsilon=0}\\ &=\left.\sigma |\phi^\varepsilon(t,x)|^{2\sigma-2}\left(\overline{\phi^\varepsilon}(t,x)\frac{\partial}{\partial \varepsilon}\phi^\varepsilon(t,x)+\phi^\varepsilon(t,x)\frac{\partial}{\partial \varepsilon}\overline{\phi^\varepsilon}(t,x)\right)\right|_{\varepsilon=0}\\ &=\left.\sigma |\phi^\varepsilon(t,x)|^{2\sigma-2}\left(R(x) w(t,x) + R(x)\overline{w}(t,x)\right)\right|_{\varepsilon=0}\\ &=\sigma R(x)^{2\sigma-1}\left(w(t,x) + \overline{w}(t,x)\right), \end{split} $$ and assuming that $F\in C^1$, the calculation of the Gateaux derivative of $\mathfrak{F}$ goes as follows $$ \begin{split} \left.\frac{\partial}{\partial \varepsilon}\right|_{\varepsilon=0}\mathfrak{F}(t,x,\phi^\varepsilon) & =2iw_te^{it/2}-we^{it/2}+\Delta w e^{it/2} \\ & \quad+\left.\left(\phi^\varepsilon \frac{\partial}{\partial \varepsilon} \vert \phi^\varepsilon\vert^{2\sigma} + \vert \phi^\varepsilon\vert^{2\sigma}\frac{\partial}{\partial \varepsilon} \phi^\varepsilon\right)\right|_{\varepsilon=0} -F(R)Re^{it/2}\\ &= 2iw_te^{it/2}-we^{it/2}+\Delta w e^{it/2} \\ & \quad + \sigma R^{2\sigma}(w + \overline{w})e^{it/2} + R^{2\sigma}we^{it/2} -F(R)Re^{it/2}\\ &= \big[2iw_t-w+\Delta w + (\sigma+1) R^{2\sigma}w + \sigma R^{2\sigma}\overline{w} -F(R)R\big]e^{it/2}\\ \end{split} $$ As we can see, the operator $\partial_\varepsilon\mathfrak{F}$ is linear (or perhaps is better to say affine) in the arguments $w$ and $\overline{w}$, thus it is the sought for linearization, and the PDE in the paper of Weinstein is simply obtained by equating it to $0$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.