# mKdV Solitons

## Modified KdV Equation *u*_{t} + 24 *u*^{2}*u*_{x} + *u*_{xxx} = 0

Travelling wave solutions of the general mKdV equation

*u*_{t} + α

*u*^{2}*u*_{x} +β

*u*_{xxx} = 0
are non-singular only when the coefficients of
the nonlinear convective term

*u*^{2}*u*_{x}
and the linear dispersive term

*u*_{xxx}
have a relative positive sign, α/β > 0
(called the defocusing case).
Up their relative sign, these coefficients α, β can be freely changed
by a scaling transformation on the variables

*u*,

*x*,

*t*.
The conventional choice α = 24, β=1 eliminates awkward numerical factors in the expressions for single and multi soliton solutions.

In contrast to the KdV equation,
a reflection transformation

*u* → −

*u* is a symmetry
for the mKdV equation.
Every mKdV solution thereby has a mirror solution
that differs only by a reflection (i.e. an overall ± sign).

### 1-Soliton Solution

The mKdV 1-soliton solution is

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

*c* > 0 is the wave speed,

*s* = ±1 is an up/down orientation of the wave profile,
and ξ =

*x*−

*ct* is a moving coordinate.

The position of the wave peak at time

*t* is

*x* =

*ct*.

### 2-Soliton Solution

The mKdV 2-soliton solution is

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

*c*_{1} >

*c*_{2} > 0 are the wave speeds,

*s*_{1} = ±1 and

*s*_{2} = ±1
are the wave orientations,
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}
and up/down orientations

*s*_{1} = ±1 and

*s*_{2} = ±1.
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*)
characterizes the nonlinear interaction of the waves in the collision.

When the two waves have the same orientation

*s*_{1} =

*s*_{2} = ±1,
their nonlinear interaction depends on whether their speed ratio
is greater or less than a critical value

*c*_{1} /

*c*_{2} = (7 + 3√5)/2
(as determined from the condition that the convexity of

*u*(

*x,0*) vanishes).
If the speed ratio is

*c*_{1} /

*c*_{2} > (7 + 3√5)/2,
the waves interact nonlinearly by first merging at

*x* =

*t* = 0 and then splitting apart.
In this case the profile

*u*(

*x,0*) has
a single peak at

*x* = 0
and an exponentially diminishing tail as

*x* → ±

∞.

**mKdV merge-split interaction**

**ratio of fast to slow wave speeds is > (7 + 3√5)/2 ≅ 6.85**

(Click here to see the space time portrait)
If instead the speed ratio is

*c*_{1} /

*c*_{2} < (7 + 3√5)/2,
the waves interact nonlinearly by first bouncing and then
exchanging shapes and speeds at

*x* =

*t* = 0.
In this case the profile

*u*(

*x,0*) has
a double peak around

*x* = 0
and an exponentially diminishing tail as

*x* → ±

∞.

**mKdV bounce-exchange interaction**
**ratio of fast to slow wave speeds is < (7 + 3√5)/2 ≅ 6.85**

(Click here to see the space time portrait)
In contrast when the two waves have opposite orientations

*s*_{1} = −

*s*_{2} = ±1,
they have a completely different nonlinear interaction
where the slow wave gradually is first absorbed and then emitted by the fast wave.
Their profile

*u*(

*x,0*) in this case has
a central peak at

*x* = 0 plus a pair of side peaks around

*x* ≠ 0
with an exponentially diminishing tail as

*x* → ±

∞.

**mKdV absorb-emit interaction**

(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 > (7 + 3√5) / 2 ≅ 6.85**

**ratio of fast to slow wave speeds is < (7 + 3√5) / 2 ≅ 6.85**
In all three 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/√*c*_{1}) ln ( (√*c*_{1} + √*c*_{2})/(√*c*_{1} − √*c*_{2}) ) > 0
and the slower wave gets shifted backward by

Δ*x*_{2} =
(2/√*c*_{2}) ln ( (√*c*_{1} − √*c*_{2})/(√*c*_{1} + √*c*_{2}) ) > 0
where √

*c*_{1} > √

*c*_{2} > 0.
These position shifts are the same as those for the KdV 2-soliton solution.

### Multi-soliton Collisions

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