Vector-MKdV Solitons II

Vector MKdV Equation u_t + 12 |u|(|u|u)_x + u_xxx = 0

In vector notation, u = (u₁,...,u_N) is a N-component variable satisfying the vector MKdV equation

u_t + 12u⋅u u_x + 12(u⋅u_x)u + u_xxx =0
For the special case when N=2, this vector equation is equivalent to the Sasa-Satsuma-MKdV equation
u_t + 6(uu_x + 3 u_xu)u + u_xxx =0 in complex-variable notation u = u₁ + i u₂.

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 c₁ and orientation (unit vector) φ₁ overtakes a slow travelling wave with speed c₂ and orientation (unit vector) φ₂ are given by the 2-soliton solution

u(t,x)= (G₁/F) φ₁ + (G₂/F) φ₂ with G₁ = √(c₁ − c₂) √c₁ ( cosh(√c₂ ξ₂)+α√c₂ exp(−√c₁ ξ₁) )
G₂ = √(c₁ − c₂) √c₂ ( sinh(√c₁ ξ₁)+α√c₁ exp(√c₂ ξ₂) )
F= 4√c₁√c₂ α( 1 + 2α(√c₁ + √c₂) exp(√c₂ ξ₂ − √c₁ ξ₁) ) + (√c₁ − √c₂) cosh(√c₁ ξ₁ + √c₂ ξ₂) + (√c₁ + √c₂) cosh(√c₁ ξ₁ − √c₂ ξ₂)
in terms of α = φ₁⋅φ₂/(√c₁ − √c₂) where ξ₁ = x−c₁t, ξ₂ = x−c₂t are moving coordinates.

Interaction Properties

Depending on their speed ratio c₁/c₂ and relative orientation angle Δφ = arccos(φ₁⋅φ₂), 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 and speed of the initial two waves. As remnants of the interaction, firstly there is a shift in the position of each wave. The faster wave is shifted forward by

Δx₁ = (1/√c₁) ln ( √(1 + 4 √c₁ √c₂ α²) (√c₁ + √c₂)/(√c₁ − √c₂) ) > 0
the slower wave is shifted backward by

Δx₂ = −(1/√c₂) ln ( √(1 + 4 √c₁ √c₂ α²) (√c₁ + √c₂)/(√c₁ − √c₂) ) < 0
in terms of α = φ₁⋅φ₂/(√c₁ − √c₂), where these position shifts depend on the speeds and relative orientation of the waves.

Secondly there is a rotation in the orientation of both waves in the plane spanned by the unit vectors φ₁ and φ₂. The faster wave has its orientation vector rotated through the angle

ν₁ = − arctan( √c₂ sin(2Δφ) )/ ( √c₁ + √c₂ cos(2Δφ))
while the slower wave has its orientation vector rotated through the angle

ν₂ = arctan( √c₁ sin(2Δφ) )/ ( √c₂ + √c₁ cos(2Δφ))
where these rotations depend on the speeds and relative orientation of the waves.

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

Merge-Split Interaction

speed ratio: c₁/c₂ = 3
relative orientation angle: Δφ = 0.6 π


speed ratio: c₁/c₂ = 50
relative orientation angle: Δφ = 0.35 π

Bounce-Exchange Interaction

speed ratio: c₁/c₂ = 3
relative orientation angle: Δφ = 0.35 π

Absorb-Emit interaction

speed ratio: c₁/c₂ = 3
relative orientation angle: Δφ = 0.85 π


speed ratio: c₁/c₂ = 50
relative orientation angle: Δφ = 0.85 π