# Vector-MKdV Solitons I

## Vector MKdV Equation *u*_{t} + 24|*u*|^{2}*u*_{x} + *u*_{xxx} = 0

In vector notation,

*u* = (

*u*_{1},...,

*u*_{N})
is a N-component variable satisfying the vector MKdV equation

*u*_{t}
+ 24 *u*⋅*u* *u*_{x}
+ *u*_{xxx} =0
For the special case when N=2,
this vector equation is equivalent to the Hirota-MKdV equation

*u*_{t}
+ 24 *u**u**u*_{x}
+ *u*_{xxx} =0
in complex-variable notation

*u* =

*u*_{1} + i

*u*_{2}.

### 1-Soliton Solution

Travelling waves are given by the 1-soliton solution

*u*(*t,x*)= (*G*/*F*) **φ**
with

*G*= √*c*, *F*= 2cosh(√*c* ξ)
where

*c* > 0 is the speed,

**φ** is the orientation unit vector,
and ξ =

*x*−

*ct* is a moving coordinate.

### 2-Soliton Solution

Collisions where a fast travelling wave with speed

*c*_{1} and orientation (unit vector)

** φ**_{1}
overtakes a slow travelling wave with speed

*c*_{2} and orientation (unit vector)

** φ**_{2}
are given by the 2-soliton solution

*u*(*t,x*)=
(*G*_{1}/*F*) **φ**_{1} + (*G*_{2}/*F*) **φ**_{2}
with

*G*_{1} =(*c*_{1} − *c*_{2})
√*c*_{1} cosh(√*c*_{2} ξ_{2}), *G*_{2} =(*c*_{1} − *c*_{2})
√*c*_{2} cosh(√*c*_{1} ξ_{1})

*F*= 4√*c*_{1}√*c*_{2} **φ**_{1}⋅**φ**_{2}
+ (√*c*_{1} − √*c*_{2})^{2}
cosh(√*c*_{1} ξ_{1} + √*c*_{2} ξ_{2})
+ (√*c*_{1} + √*c*_{2})^{2}
cosh(√*c*_{1} ξ_{1} − √*c*_{2} ξ_{2})
where ξ

_{1} =

*x*−

*c*_{1}*t*,
ξ

_{2} =

*x*−

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

### Interaction Properties

Depending on their speed ratio

*c*_{1}/*c*_{2}
and relative orientation angle Δφ = arccos(

**φ**_{1}⋅

**φ**_{2}),
the fast and slow waves in a collision exhibit three different types of nonlinear interactions.

1) Merge-Split: the waves first merge together at

*t*=

*x*=0 and then split apart into their original shapes and speeds.

2) Bounce-Exchange: the waves first bounce and then exchange their shapes and
speeds at

*t*=

*x*=0.

3) Absorb-Emit: the fast wave gradually first absorbs and then emits
the slow wave.

In all three types of interactions, the waves emerging from the collision
retain the shape, speed, and orientation of the initial two waves.
The only remnant of the interaction is a forward shift in the position of the faster wave

Δ*x*_{1} =
(2/√*c*_{1}) ln ( (√*c*_{1} + √*c*_{2})/(√*c*_{1} − √*c*_{2}) ) > 0
and a backward shift in the position of the slower wave

Δ*x*_{2} =
(2/√*c*_{2}) ln ( (√*c*_{1} − √*c*_{2})/(√*c*_{1} + √*c*_{2}) ) < 0
where these position shifts depend only on the speeds of the waves but not on their orientations.

### Overlay of 1-Soliton Solutions and Corresponding 2-Soliton Solution

#### Merge-Split Interaction

**speed ratio: ***c*_{1}/*c*_{2} = 2.8

**relative orientation angle: Δφ = 0.75 π**

**speed ratio: ***c*_{1}/*c*_{2} = 50

**relative orientation angle: Δφ = 0.3 π**
#### Bounce-Exchange Interaction

**
**

**speed ratio: ***c*_{1}/*c*_{2} = 2.8

**relative orientation angle: Δφ = 0.3 π**
**
**

#### Absorb-Emit interaction

**speed ratio: ***c*_{1}/*c*_{2} = 2.8

**relative orientation angle: Δφ = 0.95 π**

**speed ratio: ***c*_{1}/*c*_{2} = 50

**relative orientation angle: Δφ = 0.75 π**

**speed ratio: ***c*_{1}/*c*_{2} = 9

**relative orientation angle: Δφ = π**