mKdV Solitons

Modified KdV Equation u_t + 24 u²u_x + u_xxx = 0

Travelling wave solutions of the general mKdV equation u_t + αu²u_x +βu_xxx = 0 are non-singular only when the coefficients of the nonlinear convective term u²u_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)= (c₁ − c₂)( s₁√c₁ cosh(√c₂ ξ₂) + s₂√c₂ cosh(√c₁ ξ₁) )/( 4 s₁s₂√c₁√c₂
+ (√c₁ − √c₂)² cosh(√c₁ ξ₁ + √c₂ ξ₂) + (√c₁ + √c₂)² cosh(√c₁ ξ₁ − √c₂ ξ₂) )
where c₁ > c₂ > 0 are the wave speeds, s₁ = ±1 and s₂ = ±1 are the wave orientations, 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 and up/down orientations s₁ = ±1 and s₂ = ±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 s₁ = s₂ = ±1, their nonlinear interaction depends on whether their speed ratio is greater or less than a critical value c₁ / c₂ = (7 + 3√5)/2 (as determined from the condition that the convexity of u(x,0) vanishes). If the speed ratio is c₁ / c₂ > (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 c₁ / c₂ < (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 s₁ = −s₂ = ±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

Δx₁ = (2/√c₁) ln ( (√c₁ + √c₂)/(√c₁ − √c₂) ) > 0
and the slower wave gets shifted backward by

Δx₂ = (2/√c₂) ln ( (√c₁ − √c₂)/(√c₁ + √c₂) ) > 0
where √c₁ > √c₂ > 0. These position shifts are the same as those for the KdV 2-soliton solution.

Multi-soliton Collisions