 If  $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$, where $m$ and $n$ are positive integers, what is the value of $m+n$?
We start with the given equation:
\[
\frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} = \sqrt{m} - \sqrt{n}
\]

1. **Square both sides**:
\[
\left( \frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} \right)^2 = (\sqrt{m} - \sqrt{n})^2
\]
\[
\frac{1}{2011 + \sqrt{2011^2 - 1}} = m - 2\sqrt{mn} + n
\]

2. **Rationalize the denominator on the left-hand side**:
Multiply the numerator and the denominator by the conjugate of the denominator:
\[
\frac{1}{2011 + \sqrt{2011^2 - 1}} \cdot \frac{2011 - \sqrt{2011^2 - 1}}{2011 - \sqrt{2011^2 - 1}} = \frac{2011 - \sqrt{2011^2 - 1}}{(2011 + \sqrt{2011^2 - 1})(2011 - \sqrt{2011^2 - 1})}
\]

3. **Simplify the denominator**:
\[
(2011 + \sqrt{2011^2 - 1})(2011 - \sqrt{2011^2 - 1}) = 2011^2 - (\sqrt{2011^2 - 1})^2 = 2011^2 - (2011^2 - 1) = 1
\]
Thus, the expression simplifies to:
\[
\frac{2011 - \sqrt{2011^2 - 1}}{1} = 2011 - \sqrt{2011^2 - 1}
\]

4. **Equate the simplified expression to the right-hand side**:
\[
2011 - \sqrt{2011^2 - 1} = m - 2\sqrt{mn} + n
\]

5. **Separate the rational and irrational parts**:
For the equation to hold, the rational parts and the irrational parts must be equal. Therefore:
\[
m + n = 2011
\]
and
\[
-2\sqrt{mn} = -\sqrt{2011^2 - 1}
\]

6. **Solve for \(mn\)**:
\[
2\sqrt{mn} = \sqrt{2011^2 - 1}
\]
Square both sides:
\[
4mn = 2011^2 - 1
\]

7. **Verify the values of \(m\) and \(n\)**:
We need to find \(m\) and \(n\) such that:
\[
m + n = 2011
\]
\[
4mn = 2011^2 - 1
\]

By solving these equations, we can find the values of \(m\) and \(n\). However, the problem only asks for \(m + n\), which we have already determined to be:
\[
\boxed{2011}
\]