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 = u₁ + i u₂ satisfies

u_t + 6(uu_x + 3 u_xu)u + u_xxx =0
(2) In vector notation, u = (u₁, u₂) 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 c₁ and phase φ₁ overtakes a slow travelling wave with speed c₂ and phase φ₂ are given by the special 2-soliton solution

u(t,x)= (c₁ − c₂)² (√σ/2) ( √c₁ exp(iφ₁) ( exp(iν₁ + √c₂ ξ₂) + exp(−√c₂ ξ₂) ) + √c₂ exp(iφ₂) ( exp(iν₂ − √c₁ ξ₁) + exp(√c₁ ξ₁) ) )/( 4 √c₁√c₂ cos(φ₁ − φ₂) + (√c₁ − √c₂)² cosh(√c₁ ξ₁ + √c₂ ξ₂) + (√c₁ + √c₂)²σ cosh(√c₁ ξ₁ − √c₂ ξ₂) )
with σ = √( c₁ + c₂ +2 √c₁ √c₂ cos(2(φ₁ − φ₂)) )/( √c₁ + √c₂ ) and ν₁ = arctan( √c₂ sin(2(φ₁ − φ₂)) )/( √c₁ + √c₂ cos(2(φ₁ − φ₂)) )
ν₂ = arctan( √c₁ sin(2(φ₂ − φ₁)) )/( √c₂ + √c₁ cos(2(φ₂ − φ₁)) )
where ξ₁ = x−c₁t, ξ₂ = x−c₂t 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 c₁/c₂ = 3
difference in phases is |φ₁ − φ₂| = 0.6 π
(Click here to see the space time portrait)

Sasa-Satsuma bounce-exchange interaction ratio of fast to slow wave speeds is c₁/c₂ = 3
difference in phases is |φ₁ − φ₂| = 0.35 π
(Click here to see the space time portrait)


Sasa-Satsuma critical-phase interaction ratio of fast to slow wave speeds is c₁/c₂ = 3
difference in phases is |φ₁ − φ₂| = 0.70316 π
(Click here to see the space time portrait)

Sasa-Satsuma absorb-emit interaction
ratio of fast to slow wave speeds is c₁/c₂ = 3
difference in phases is |φ₁ − φ₂| = 0.85 π
(Click here to see the space time portrait)

Sasa-Satsuma merge-split interaction ratio of fast to slow wave speeds is c₁/c₂ = 50
difference in phases is |φ₁ − φ₂| = 0.35 π
(Click here to see the space time portrait)

Sasa-Satsuma critical-phase interaction ratio of fast to slow wave speeds is c₁/c₂ = 50
difference in phases is |φ₁ − φ₂| = 0.6273 π
(Click here to see the space time portrait)

Sasa-Satsuma absorb-emit interaction ratio of fast to slow wave speeds is c₁/c₂ = 50
difference in phases is |φ₁ − φ₂| = 0.85 π
(Click here to see the space time portrait)