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Robin04
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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?
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?
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