# Generalized Virial Identity for NLS

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## Latest revision as of 16:15, 27 May 2010

Linking back to the course: 2010 MAT495H1 Group: Solitons and Blowup. (page under development)

# The Generalized Nonlinear Schrodinger

We consider the Cauchy problem for the Generalized Nonlinear Schrodinger equation

$$(i\partial_t + \Delta)u =\pm F'(|u|^{2})u$$ with initial condition $$u(0,x)=u_{o}(x).$$ We have that $F'(x)\ge 0$. In the focusing case we have a $+$ sign and defocusing a $-$ sign. This can be specialized into several specific NLS equations, but we will use this general case to note the conserved quantities.

# Conserved Quantities

We can show that mass, momentum, and energy are conserved. Specifically, $$\frac{\partial}{\partial t}\int_{\mathbb{R}^{n}}|u(t,x)|^{2}dx=0,$$

$$\frac{\partial}{\partial t}2\Im\int_{\mathbb{R}^{n}} \bar{u}(t)\nabla u(t)dx=0,$$

and

$$\frac{\partial}{\partial t}H[u(t)]=\frac{\partial}{\partial t}\frac{1}{2}\int_{\mathbb{R}^{n}} |\nabla u(t)|^{2}\pm F(|u(t)|^{2})=0.$$

We will not show these conserved quantities but they are a simple manipulations of the NLS equation. We introduce some notation and physical interpretation to make our the rest of what is written easier to read and understand. Imagine the solution $u$ as a particle, the mass as $$T_{oo}:=|u|^{2},$$ the momentum as $$T_{oj}:=T_{jo}=2\Im (u\bar{u}_j),$$ and the momentum current as

$$T_{jk}:=T_{kj}=4\Re(u_k\bar{u}_j)-\delta_{j,k}\Delta(|u|^{2})+2\delta_{j,k}\sigma(|u|^2).$$ Where $\sigma(z):=zF'(z)-F(z)\sim F(z)$.

From these definitions, one can show by computation that $$\partial_{t}T_{oo}=\partial_{x_{j}}T_{oj}$$ and $$\partial_{t}T_{ko}=\partial_{x_{j}}T_{kj}.$$ Note, we use Einstein notation here.

The first equation comes from

\begin{align} \partial_t (u\bar{u})&=u_t\bar{u}+u\bar{u}_t\\ &=-i(iu_t\bar{u})+c.c.\\ &=-i(-\Delta u+F'(|u|^2)u)\bar{u}+c.c.\\ &=i(\Delta u)\bar{u}+c.c. \end{align}

and

\begin{align} \partial_j[2Im(u \bar{u}_j)]&=(-i)\partial_j(u\bar{u}_j-\bar{u}u_j)\\ &=(-i)(u_j\bar{u}_j+u\bar{u}_{jj}-\bar{u}_ju_j-\bar{u}u_{jj})\\ &=i(\bar{u}u_{jj}-u\bar{u}_{jj}) \end{align}

where $c.c.$ means complex conjuate of the previous terms.

The LHS of the second equation is

\begin{align} \partial_t[2\Im(u\bar{u}_k)]&=(-i)\partial_t[u\bar{u}_k-c.c]\\ &=(-i)[u_t\bar{u}_k+u\bar{u}_{kt}-c.c.]\\ &=-[(iu_t)\bar{u}_k-u\overline{(iu)}_{kt}]+c.c.\\ &=-[(-\Delta u+F'(|u|^2)u)\bar{u_k}-u\overline{(-\Delta u+F'(|u|^2)u)}_k]+c.c.\\ &=(\Delta u)\bar{u}_k-(\Delta\bar{u}_k)u-F'(|u|^2)u\bar{u}_k+[F'(|u|^2)\bar{u}]_ku+c.c. \end{align}

and because

$$\Delta(|u|^2)_k=[(\Delta u)\bar{u}+2\nabla u\cdot\nabla\bar{u}+u(\Delta\bar{u})]_k,$$

the RHS of the second equation is

\begin{align} \partial_j T_{kj}&=\partial_j[2(u_k\bar{u}_j+\bar{u}_k u_j)-\delta_{jk}\Delta(|u|^2)+2\delta_{jk}\sigma(|u|^2)]\\ &=(\Delta\bar{u})u_k+(\Delta u)\bar{u}_k-(\Delta u_k)\bar{u}-u(\Delta \bar{u}_k)+[|u|^2F'(|u|^2)-F(|u|^2)]_k, \end{align}

hence the second equation holds.

# Generalized Virial Identity

We now introduce the the the Virial potential associated to $a$, $$V_{o}^{a}(t):=\int_{\mathbb{R}^{d}} a(t,x) |u|^{2}(t,x)dx=\int_{\mathbb{R}^{d}} a(t,x) T_{oo}dx,$$ and the Morawetz action $$M_{o}^{a}(t):=\partial_t V_{o}^{a}(t)=\int_{\mathbb{R}^{d}} a_{t}T_{oo}+a\partial_{x_{j}}T_{oj}dx=\int_{\mathbb{R}^{d}} a_{t}T_{oo}-a_{x_{j}}T_{oj}dx.$$ The virial identity is derived from the two equalities show in the previous section.

\begin{align} \partial_t^2V_0^a(t)=\partial_tM_0^a&=\int_{\mathbb{R}^{d}} a_{tt}T_{00t}+a_tT_{00} dx-\int_{\mathbb{R}^{d}}a_{tj}T_{0j}+a_jT_{0jt}dx\\ &=\int_{\mathbb{R}^{d}}a_{tt}|u|^2+a_{t}\partial_jT_{0j}dx-\int_{\mathbb{R}^{d}}a_{tj}T_{0j}+a_j\partial_kT_{jk}dx\\ &=\int_{\mathbb{R}^{d}}a_{tt}|u|^2dx-\int_{\mathbb{R}^{d}}2a_{tj}T_{0j}+a_j\partial_kT_{jk}dx\\ &=\int_{\mathbb{R}^{d}}a_{tt}|u|^2dx-4\int_{\mathbb{R}^{d}}a_{tj}\Im(u\bar{u}_j)-a_{jk}T_{jk}dx\\ &=\int_{\mathbb{R}^{d}}(a_{tt}-\Delta\Delta a)|u|^2dx-4\int_{\mathbb{R}^{d}}a_{tj}\Im(u\bar{u}_j)dx+4\int_{\mathbb{R}^{d}} a_{jk}\Re(u_k\bar{u}_j)dx\\ & +2\int_{\mathbb{R}^{d}}\Delta a\sigma(|u|^2)dx \end{align}

# Example One: Virial Identity

If we take the weight $a(x)=|x|^{2}$, $a_{jk}=2\delta_{jk}$, $\nabla a=2d$, $\nabla\nabla a=0$, to arrive at $$\partial_{t}^{2}V_{o}^{a}(t)=+4\int_{\mathbb{R}^{d}} \delta_{jk}\Re(u_k\bar{u}_j)dx+4d\int_{\mathbb{R}^{d}}\sigma(|u|^2)dx$$ $$=8H+(\le 0)\le8h<0$$ since we are assuming the focusing case and specific initial data with negative initial (and therefore all-time) energy. We have shown $$\partial_{t}^{2} \int_{\mathbb{R}^{d}} a(t,x) |u|^{2}(t,x)dx<0$$ which tells us that $\int_{\mathbb{R}^{d}} a(t,x) |u|^{2}(t,x)dx$ is positive and monotone decreasing, telling us that at the point at which this equals zero, say $t=T^\star$ we have nonexistance of solution. $T^\star$ is the blow up time in this case. This is the argument which is found in Glassey's paper.

To follow.