# Hirota Solitons

## Hirota-mKdV Equation *u*_{t} + 24|*u*|^{2}*u*_{x} + *u*_{xxx} = 0

There are two equivalent forms for the Hirota-mKdV equation:

(1) In complex-variable notation,

*u* =

*u*_{1} + i

*u*_{2} satisfies

*u*_{t}
+ 24 *u**u**u*_{x}
+ *u*_{xxx} =0
(2) In vector notation,

*u* = (

*u*_{1},

*u*_{2}) satisfies

*u*_{t}
+ 24 *u*⋅*u* *u*_{x}
+ *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})(
√*c*_{1} exp(iφ_{1})
cosh(√*c*_{2} ξ_{2})
+ √*c*_{2} exp(iφ_{2})
cosh(√*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}
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.

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

**Hirota merge-split interaction**

**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 2.8

**difference in phases is |φ**_{1} − φ_{2}| = 0.75 π

(Click here to see the space time portrait)
**Hirota bounce-exchange interaction**
**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 2.8

**difference in phases is |φ**_{1} − φ_{2}| = 0.3 π

(Click here to see the space time portrait)

**Hirota critical-phase interaction**
**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 2.8

**difference in phases is |φ**_{1} − φ_{2}| = 0.8689 π

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

**ratio of fast to slow wave speeds is ***c*_{1}/*c*_{2} = 2.8

**difference in phases is |φ**_{1} − φ_{2}| = 0.95 π

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

**difference in phases is |φ**_{1} − φ_{2}| = 0.3 π

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

**difference in phases is |φ**_{1} − φ_{2}| = 0.637 π

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

**difference in phases is |φ**_{1} − φ_{2}| = 0.75 π

(Click here to see the space time portrait)