KdV Solitons

KdV Equation u_t + αuu_x + βu_xxx = 0

The coefficients α, β in the general form of the KdV equation can be freely changed through scaling/reflection transformations on the variables u, x, t. A conventional choice is α = 6, β = 1, which eliminates awkward numerical factors in the expressions for soliton solutions.

1-Soliton Solution

The KdV 1-soliton solution is
u(x,t)= (c/2)/cosh²(√c ξ/2)
where c > 0 is the wave speed and ξ = x−ct is a moving coordinate. The position of the wave peak at time t is x = ct.

2-Soliton Solution

The KdV 2-soliton solution is
u(x,t)= 2(c₁ − c₂) ( c₁ cosh²(√c₂ ξ₂/2) + c₂ sinh²(√c₁ ξ₁/2) )/( (√c₁ − √c₂) cosh((√c₁ ξ₁ + √c₂ ξ₂)/2)
+ (√c₁ + √c₂) cosh((√c₁ ξ₁ − √c₂ ξ₂)/2) )²
where c₁ > c₂ > 0 are the wave speeds and ξ₁ = x−c₁t, ξ₂ = x−c₂t 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 c₁ and c₂. 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) is characterized by its convexity u(x,0)_xx|_x=0 which is positive or negative depending on whether the speed ratio is greater or less than a critical value c₁ / c₂ = 3 (for which the convexity vanishes).

For speed ratios c₁ / c₂ > 3, the profile has a single peak at x = 0 and an exponentially diminishing tail as x → ±∞. In this case the waves interact nonlinearly by first merging at x = t = 0 and then splitting apart. KdV merge-split interaction
ratio of fast to slow wave speeds is > 3
(Click here to see the space time portrait)

For speed ratios c₁ / c₂ < 3, the profile has a double peak around x = 0 and an exponentially diminishing tail as x → ±∞. In this case the waves interact nonlinearly by first bouncing and then exchanging shapes and speeds at x = t = 0. KdV bounce-exchange interaction
ratio of fast to slow wave speeds is < 3
(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 > 3
ratio of fast to slow wave speeds is < 3 In both 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
Δx₁ = (2/k₁) ln ( (k₁ + k₂)/(k₁ − k₂) ) > 0
and the slower wave gets shifted backward by
Δx₂ = (2/k₂) ln ( (k₁ − k₂)/(k₁ + k₂) ) < 0
where k₁ = √c₁ > k₂ = √c₂ > 0.

Multi-soliton Collisions

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

3-Soliton Simultaneous Collision Solution

(Click here to see the space time portrait)
All three waves collide at x = 0, t = 0.

3-Soliton Pair-wise Collision Solution

(Click here to see the space time portrait)
The two fastest waves interact first and then overtake the slowest wave

3-Soliton Pair-wise Collision Solution

(Click here to see the space time portrait)
The two slowest waves interact and eventually both are overtaken by the fastest wave.