Understanding the definition of a soliton

[Robin04]

I'm learning about solitons from a book called Solitons and Instantons by R. Rajaraman.

He defines (page 14-15) a soliton as a solution to a (possibly non-linear) PDE where the energy density of the system is of the form $\epsilon (x,t) = \sum_i \epsilon_0(x-a_i-u_i t-\delta_i)$, as $t \rightarrow \infty$ where the i index permits that we have more waves traveling with speed $u_i$, $a_i$ is their initial positions, and $\epsilon_0$ is the energy density resulting from a single wave.

My problem is with understanding what $\delta_i$ means. Here's what the books says:
"$\delta_i$ represents the possibility that the solitons may suffer a bodily displacement compared with their pre-collision trajectories. This displacement should be the sole residual effect of the collisions if they are to be solitons."

The picture I have in mind about solitons is that after collision they look like if they hadn't collided, so why do we need this extra displacement?

 

[jedishrfu]

Could this be due to the soliton being formed some short distance from the wave emitter. As an example, a boat traveling down a canal pushes water in front of it and then stops abruptly and the water waves continue traveling. The center of the wave packet isn’t at the boats prow necessarily right?

 

[Ibix]

The picture I have in mind about solitons is that after collision they look like if they hadn't collided, so why do we need this extra displacement?

Isn't it just saying that if you run alongside a soliton (at speed $u_i$) and it interacts with another soliton you may, afterwards, find that you are ahead or behind your soliton by distance $\delta_i$? I wouldn't regard a wholesale displacement as a change to a soliton - otherwise it could only be frozen in place.

I suspect you know more about solitons than I do if you are actually studying them, but generally you should take verbal descriptions of physical phenomena (like "solitons don't change") with a grain of salt. Trust the maths. If there are solutions to the PDEs that have the stated property then such solutions exist, whether you call them solitons or not, and whether you describe them as "not changing" or not. Your author apparently does call them solitons. Whether this is common practice or not I couldn't say.

 

[Robin04]

Could this be due to the soliton being formed some short distance from the wave emitter. As an example, a boat traveling down a canal pushes water in front of it and then stops abruptly and the water waves continue traveling. The center of the wave packet isn’t at the boats prow necessarily right?

I wasn't precise enough with the definition. It has another part: $\epsilon = \sum_i \epsilon_0(x-a_i-u_it)$, as $t \rightarrow -\infty$
So this $\delta_i$ has to come from the collision.

Isn't it just saying that if you run alongside a soliton (at speed $u_i$) and it interacts with another soliton you may, afterwards, find that you are ahead or behind your soliton by distance $\delta_i$? I wouldn't regard a wholesale displacement as a change to a soliton - otherwise it could only be frozen in place.

I suspect you know more about solitons than I do if you are actually studying them, but generally you should take verbal descriptions of physical phenomena (like "solitons don't change") with a grain of salt. Trust the maths. If there are solutions to the PDEs that have the stated property then such solutions exist, whether you call them solitons or not, and whether you describe them as "not changing" or not. Your author apparently does call them solitons. Whether this is common practice or not I couldn't say.

You're right. Probably this extra displacement is included only to allow certain non-linear equations' solutions to be called solitons as well, as there quite a few. For linear equations $\delta_i=0$ for sure, and my intuition is still too limited to linear equations but coming back to the problem after a few days, now I see that this extra term doesn't say much about the physics of solitons. And indeed, the author mentioned that there's not really a commonly accepted definition, some solutions are called solitons according to one definition but excluded by others.