What is the solution of $x^a+(1+x)^b=0 $? I am not a mathematician, but a theoretical physicist. I am faced with this equation coming from some plasma phenomenon and  I am unable to 'recognise' it. 
Mathematica software cannot solve it. I have searched the web and it might have relation with the Lambert function, but I still can't clearly see how.
I apologise before hand if this is too trivial a question or perhaps too difficult, but would enormously appreciate any hint. Many thanks. Cheers. $($I am not even sure under what TAG to classify this question... :-( $)$ 
To make it clearer $($sorry for not having mentioned it$)$: $a$ and $b$ are real positive numbers and $x$ $($which is actually a function$)$ has real values.

Assuming (as per comments) that $a$ and $b$ are actually positive integers then we have four cases:
1) $a$ even, $b$ even - no real solution.
2) $a$ even, $b$ odd - one real solution in $(-\infty, -2)$ if $a<b$, otherwise no real solutions.
3) $a$ odd, $b$ even - one real solution in $(-1,0)$.
4) $a$ odd, $b$ odd - one real solution in $(-1,0)$ - special case  if $a=b$ then solution is $x=-\frac12$.
Where a real solution exists it shouldn't be difficult to find a good numerical approximation using Newton's method.
