9
$\begingroup$

I'm reading [1] recently and have another question about a remark in this paper. I tried to solve it myself (see below) but did not succeed. It could be just a notation problem.

The Setup:

  • Let $u \in H^1(\Omega,\mathbb{C})$ where $\Omega = [-\pi,\pi]^3 \subset \mathbb{R}^3$.

  • In [1, Rmk.4.2] the authors denote by $Ju$ the 2-form \begin{equation} Ju \equiv \frac{1}{2} d(u \times du) = \sum_{1 \leq i < j \leq 3} (\partial_i u \times \partial_j u) dx_i \wedge dx_j. \end{equation}

  • Furthermore they define \begin{equation} \zeta_1(x)= - x_2 dx_1 \wedge dx_2 - x_3 dx_1 \wedge dx_3. \end{equation}

  • From this definition it follows that \begin{equation} \star \zeta_1 = -x_2 dx_3 + x_3 dx_2 \end{equation} which I checked. Here $\star$ denotes the Hodge-star-operator.

  • I then calculated \begin{equation} d(\star \zeta) = -2dx_2 \wedge dx_3 \tag{G1}\end{equation} which I hope is correct.

  • Now the authors state that \begin{equation} (u \times du) \wedge d(\star \zeta_1) = 2 \langle i \partial_1 u, u \rangle dx_1 \wedge \ldots \wedge dx_3 \tag{G2} \end{equation}

My Question:

What is $u \times du$ supposed to be?

My Attempt:

I tried to find out myself, but I discovered the following difficulty:

  • Let $\omega=\sum_{i=1}^3 \omega^i dx_i$ be the 1-Form $\omega=u \times du$. Then \begin{equation} \frac{1}{2} d\omega = \frac{1}{2} \sum_{1 \leq i< j \leq 3} \omega^i_{x_j} dx_j \wedge dx_i\stackrel{!}{=} Ju = \sum_{1\leq i < j\leq 3} (\partial_i u \times \partial_j u) dx_i \wedge dx_j. \end{equation} This suggests that $\omega_{x_j}^i = -2 (\partial_i u \times \partial_j u)$.

  • On the other hand because of $(G1)$ equation $(G2)$ becomes \begin{equation} \omega \wedge d(\star \zeta) = -2\omega^1dx_1 \wedge dx_2 \wedge dx_3 \stackrel{!}{=} 2 \langle i \partial_1 u, u \rangle dx_1 \wedge dx_2 \wedge dx_3, \end{equation} which suggests $\omega^1=-\langle i \partial_1 u, u \rangle$.

So my question reduces to

How are these two equations compatible? What is the $\times$ supposed to mean?


[1] Béthuel, F., P. Gravejat und J. C. Saut: Travelling waves for the Gross- Pitaevskii equation. II. Comm. Math. Phys., 285(2):567–651, 2009.

$\endgroup$
5
  • $\begingroup$ Isn't it simply $u\times du = u\, du$? $\endgroup$ Commented Jul 3, 2013 at 14:11
  • $\begingroup$ Well if that was true, i.e. when $\times = \cdot$ then $d(udu)=d(\sum u u_{x_i} dx_i)=\sum u_{x_j} u_{x_i}+u u_{x_i x_j} dx_j \wedge dx_i \neq \sum u_{x_i} u_{x_j} dx_i \wedge dx_j$, right? $\endgroup$
    – mjb
    Commented Jul 3, 2013 at 15:09
  • $\begingroup$ $d(u\,du)=0$, so no help there. $\endgroup$ Commented Jul 3, 2013 at 15:20
  • $\begingroup$ What does $H^1$ signify? $\endgroup$
    – Muphrid
    Commented Jul 3, 2013 at 16:39
  • $\begingroup$ @Muphrid The Sobolev space $W^{1,2}$ $\endgroup$
    – mjb
    Commented Jul 4, 2013 at 6:57

1 Answer 1

7
$\begingroup$

I believe this is all resolved if you have one typo. If $u$ maps $\Omega$ to $\mathbb R^3$, then you are taking the cross product of a vector in $\mathbb R^3$ with an $\mathbb R^3$-valued $1$-form. This is totally consistent with their formulas.

So, we both win.:) They are interpreting $\mathbb C \cong \mathbb R^2$ and defining the cross product of two vectors $u,v\in\mathbb R^2$ as the real number $(u\times v)\cdot e_3\in\mathbb R$. In particular, $$u\times du = \big(u_2(\partial_1u_1) - u_1(\partial_1 u_2)\big)dx_1 \pmod{dx_2,dx_3}\,.$$

All their formulas are consistent with this, recalling that they've defined $\langle u,v\rangle = \text{Re}(u\bar v)$, i.e., the real dot product of the vectors in $\mathbb C \cong \mathbb R^2$.

$\endgroup$
5
  • 1
    $\begingroup$ That may be true but as you can see here www.math.polytechnique.fr/~gravejat/Recherch/BGS1-GP.pdf on page 44, this is not a typo. I need $u$ to be in $H^1(\Omega,\mathbb{C})$ $\endgroup$
    – mjb
    Commented Jul 3, 2013 at 15:01
  • $\begingroup$ Thank you very much! I am still struggling to understand your post: What do you mean by $u \times v$ if $u,v \in \mathbb{R}^2$? and why is $(u \times v) \cdot e_3 \in \mathbb{R}$? And most important of all: what do you mean by $\mbox{mod } dx_2, dx_3$? Sorry for these questions, but I'm not very familiar with differential forms. Thanks in advance! $\endgroup$
    – mjb
    Commented Jul 4, 2013 at 6:54
  • 2
    $\begingroup$ Oh, I meant that you can take the cross product of two vectors in $\mathbb R^2$ by thinking if them as vectors in $\mathbb R^3$ with their last coordinate equal to $0$. the answer is a vector of the form $(0,0,\#)$; by taking just $\#$ we think of this as a real number. Next, since you are going to wedge your form with $dx_2\wedge dx_3$, any terms with $dx_2$ or $dx_3$ will then disappear, so by saying "mod" I meant we'll ignore those terms. $\endgroup$ Commented Jul 4, 2013 at 11:42
  • $\begingroup$ Thanks a lot! That indeed explains equation (G2)! But how can I calculate $d(u \times du)$ from that? If $u \times du = (u_2 \partial_1 u_1 - u_1 \partial_1 u_2)dx_1 + \ldots$, then I would get mixed derivatives in $d(u \times du)$, right? For example $\partial_2 (u_2 \partial_1 u_1) = \partial_2 u_2 \partial_1 u_1 + u_2 \partial_2 \partial_1 u_1 $. But in the definition of $d(u \times du)$ I don't see such mixed derivatives! Where is my mistake? $\endgroup$
    – mjb
    Commented Jul 4, 2013 at 14:25
  • 1
    $\begingroup$ Well, for that, we can't ignore the $dx_2$ and $dx_3$ terms. By the product rule, it is $du\times du$ and you get the authors' formula in your first displayed equation. $$du\times du = \big(\sum \partial_ku \,dx_k\big)\times \big(\sum \partial_ju \,dx_j\big)\,.$$ For example, the $dx_1\wedge dx_2$ coefficient will be $\partial_1(u_1+iu_2)\times\partial_2(u_1+iu_2)$. Work it out. $\endgroup$ Commented Jul 4, 2013 at 14:53

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .