# Sasa-Satsuma Solitons

## Sasa-Satsuma-mKdV Equation *u*_{t} + 12 |*u*|(|*u*|*u*)_{x} + *u*_{xxx} = 0

There are two equivalent forms for the Sasa-Satsuma-mKdV equation:

(1) In complex-variable notation,

*u* =

*u*_{1} + i

*u*_{2} satisfies

*u*_{t}
+ 6(*u**u*_{x} + 3 *u*_{x}*u*)*u*
+ *u*_{xxx} =0
(2) In vector notation,

*u* = (

*u*_{1},

*u*_{2}) satisfies

*u*_{t}
+ 12(*u*⋅*u* *u*_{x} + (*u*⋅*u*_{x})*u*)
+ *u*_{xxx} =0
### Travelling Wave Solution

Travelling waves are given by the special soliton solution

*u*(*t,x*)= (√*c* /2) exp(iφ)/cosh(√*c* ξ)
where

*c* > 0 is the speed,
−π ≤ φ ≤ π is the phase,
and ξ =

*x*−

*ct* is a moving coordinate.

### Colliding Travelling-Waves Solution

Collisions where a fast travelling wave with speed

*c*_{1} and phase φ

_{1}
overtakes a slow travelling wave with speed

*c*_{2} and phase φ

_{2}
are given by the special 2-soliton solution

*u*(*t,x*)=
(*c*_{1} − *c*_{2})^{2} (√σ/2) (
√*c*_{1} exp(iφ_{1}) (
exp(iν_{1} + √*c*_{2} ξ_{2})
+ exp(−√*c*_{2} ξ_{2}) )
+ √*c*_{2} exp(iφ_{2}) (
exp(iν_{2} − √*c*_{1} ξ_{1})
+ exp(√*c*_{1} ξ_{1}) )
)/(
4 √*c*_{1}√*c*_{2} cos(φ_{1} − φ_{2})
+ (√*c*_{1} − √*c*_{2})^{2}
cosh(√*c*_{1} ξ_{1} + √*c*_{2} ξ_{2})
+ (√*c*_{1} + √*c*_{2})^{2}σ
cosh(√*c*_{1} ξ_{1} − √*c*_{2} ξ_{2})
)
with

σ = √( *c*_{1} + *c*_{2}
+2 √*c*_{1} √*c*_{2} cos(2(φ_{1} − φ_{2}))
)/( √*c*_{1} + √*c*_{2}
)
and

ν_{1} =
arctan(
√*c*_{2} sin(2(φ_{1} − φ_{2}))
)/( √*c*_{1} + √*c*_{2} cos(2(φ_{1} − φ_{2}))
)

ν_{2} =
arctan(
√*c*_{1} sin(2(φ_{2} − φ_{1}))
)/( √*c*_{2} + √*c*_{1} cos(2(φ_{2} − φ_{1}))
)
where ξ

_{1} =

*x*−

*c*_{1}*t*,
ξ

_{2} =

*x*−

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

### Overlay of Travelling-Wave Solutions and Corresponding Collision Solution

**Sasa-Satsuma merge-split interaction**

**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 3

**difference in phases is |φ**_{1} − φ_{2}| = 0.6 π

(Click here to see the space time portrait)
**Sasa-Satsuma bounce-exchange interaction**
**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 3

**difference in phases is |φ**_{1} − φ_{2}| = 0.35 π

(Click here to see the space time portrait)

**Sasa-Satsuma critical-phase interaction**
**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 3

**difference in phases is |φ**_{1} − φ_{2}| = 0.70316 π

(Click here to see the space time portrait)
**Sasa-Satsuma absorb-emit interaction**

**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 3

**difference in phases is |φ**_{1} − φ_{2}| = 0.85 π

(Click here to see the space time portrait)
**Sasa-Satsuma merge-split interaction**
**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 50

**difference in phases is |φ**_{1} − φ_{2}| = 0.35 π

(Click here to see the space time portrait)
**Sasa-Satsuma critical-phase interaction**
**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 50

**difference in phases is |φ**_{1} − φ_{2}| = 0.6273 π

(Click here to see the space time portrait)
**Sasa-Satsuma absorb-emit interaction**
**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 50

**difference in phases is |φ**_{1} − φ_{2}| = 0.85 π

(Click here to see the space time portrait)