I've got quite easy question (and my attempt to solve it). So there is an optical fiber with Dispersion index: D = -100ps/nm•km, length: L = 10km. The question is: which wavelength will achieve faster speed and the end of the optical fibre: λ_1 = 1000nm or λ_2 = 1001nm? In my opinion if the speed of the wave is dependent on the refractive index which will be the same in the same fibre then the speed of both wavelengths will have the same speed. Is it true? Does dispersion have any impact here?
First, when you talk about the "speed" of a signal in optical fiber, that's ambiguous. You should be clear about whether you're interested in the latency (the time it takes a signal to travel from one end of the fiber to the other) or the bit rate. In this case, it seems most likely you're interested in the latency, or propagation delay.
In my opinion if the speed of the wave is dependent on the refractive index which will be the same in the same fibre then the speed of both wavelengths will have the same speed. Is it true?
No. This is not true. The index of refraction of a material varies (at least slightly) depending on the wavelength of the light being considered.
In addition, in a dielectric waveguide like optical fiber, as the wavelength changes a different proportion of the signal power travels in the core and in the cladding, leading to (at least small) changes in the effective index of the fiber.
In fact, dispersion can be either negative or positive (also called anomolous and normal dispersion), depending on the wavelength and the design of the fiber. We can also engineer the dispersion properties of the fiber in some cases to optimize the fiber for different applications.
But all of that is irrelevant to answering the question, because the total effect is summarized in the dispersion parameter.
When you specify the dispersion as you did, D = -100ps/nm•km, you're saying we already know the effect of all those variations, and that effect is that the propagation delay through 1 km of fiber changes by -100 ps for every nanometer of change in the wavelength of the signal light.
So you don't need to worry about the physical mechanism. You just need to apply the definition of the dispersion parameter to decide whether a longer or shorter wavelength travels faster through this fiber.