# KdV Solitons

## KdV Equation *u*_{t} + α*uu*_{x} + β*u*_{xxx} = 0

The coefficients α, β in the general form of the KdV equation
can be freely changed through scaling/reflection transformations
on the variables

*u*,

*x*,

*t*.
A conventional choice is α = 6, β = 1,
which eliminates awkward numerical factors in the expressions for soliton solutions.

### 1-Soliton Solution

The KdV 1-soliton solution is

*u*(*x,t*)= (*c*/2)/cosh^{2}(√*c* ξ/2)
where

*c* > 0 is the wave speed
and ξ =

*x*−

*ct* is a moving coordinate.

The position of the wave peak at time

*t* is

*x* =

*ct*.

### 2-Soliton Solution

The KdV 2-soliton solution is

*u*(*x,t*)=
2(*c*_{1} − *c*_{2})
( *c*_{1} cosh^{2}(√*c*_{2} ξ_{2}/2)
+ *c*_{2} sinh^{2}(√*c*_{1} ξ_{1}/2)
)/( (√*c*_{1} − √*c*_{2})
cosh((√*c*_{1} ξ_{1} + √*c*_{2} ξ_{2})/2)

+ (√*c*_{1} + √*c*_{2})
cosh((√*c*_{1} ξ_{1} − √*c*_{2} ξ_{2})/2)
)^{2}
where

*c*_{1} >

*c*_{2} > 0 are the wave speeds
and ξ

_{1} =

*x*−

*c*_{1}*t*,
ξ

_{2} =

*x*−

*c*_{2}*t* are moving coordinates.

As

*t* → ±

∞,
the 2-soliton solution has the asymptotic form of a linear superposition of
travelling waves (i.e. single solitons) with speeds

*c*_{1} and

*c*_{2}.
The two waves wave collide such that the moment of
greatest nonlinear interaction occurs at

*t* = 0
when the profile is symmetric is symmetric around

*x* = 0.
The shape of this profile

*u*(

*x,0*)=

*u*(

*−x,0*) is
characterized by its convexity

*u*(

*x,0*)

_{xx}|_{x=0}
which is positive or negative depending on whether the speed ratio
is greater or less than a critical value

*c*_{1} /

*c*_{2} = 3
(for which the convexity vanishes).

For speed ratios

*c*_{1} /

*c*_{2} > 3,
the profile has a single peak at

*x* = 0
and an exponentially diminishing tail as

*x* → ±

∞.
In this case the waves interact nonlinearly by first merging at

*x* =

*t* = 0 and then splitting apart.

**KdV merge-split interaction**

**ratio of fast to slow wave speeds is > 3**

(Click here to see the space time portrait)
For speed ratios

*c*_{1} /

*c*_{2} < 3,
the profile has a double peak around

*x* = 0
and an exponentially diminishing tail as

*x* → ±

∞.
In this case the waves interact nonlinearly by first bouncing and then
exchanging shapes and speeds at

*x* =

*t* = 0.

**KdV bounce-exchange interaction**

**ratio of fast to slow wave speeds is < 3**

(Click here to see the space time portrait)
### Overlay of 1-soliton solutions and corresponding 2-soliton solution

**ratio of fast to slow wave speeds is > 3**

**ratio of fast to slow wave speeds is < 3**
In both types of interactions, the waves emerging from the collision
retain the shape and speed of the initial two waves,
but the collision causes a position shift such that
the faster wave gets shifted forward by

Δ*x*_{1} =
(2/*k*_{1}) ln ( (*k*_{1} + *k*_{2})/(*k*_{1} − *k*_{2}) ) > 0
and the slower wave gets shifted backward by

Δ*x*_{2} =
(2/*k*_{2}) ln ( (*k*_{1} − *k*_{2})/(*k*_{1} + *k*_{2}) ) < 0
where

*k*_{1} = √

*c*_{1} >

*k*_{2} = √

*c*_{2} > 0.

### Multi-soliton Collisions

Multi-soliton solutions have the same interaction features displayed by the 2-soliton solution.

### 3-Soliton Simultaneous Collision Solution

(Click here to see the space time portrait)
All three waves collide at

*x* = 0,

*t* = 0.

### 3-Soliton Pair-wise Collision Solution

(Click here to see the space time portrait)
The two fastest waves interact first and then overtake the slowest wave

### 3-Soliton Pair-wise Collision Solution

(Click here to see the space time portrait)
The two slowest waves interact and eventually both are overtaken by the fastest wave.