The coarse-grain description of interacting sine-Gordon solitons with varying widths

We study the dynamics of the sine-Gordon equation's kink soliton solutions under the coarse-grain description via two "collective variables": the position of the "center" of a soliton and its characteristic width ("size"). Integral expressions for the interaction potential and the quasi-particles' cross-masses are derived. However, these cannot be evaluated in closed form when the solitons have varying widths, so we develop a perturbation approach with the velocity of the faster soliton as the small parameter. This enables us to derive a system of four coupled second-order ODEs, one for each collective variable. The resulting initial-value problem is very stiff and numerical instabilities make it difficult to solve accurately, so a semi-empirical iterative approach to its solution is proposed. Then, we demonstrate that, even though it appears the solitons pass through each other, the quasi-particles actually "exchange" their pseudomasses during a collision.

where ξ = x − vt and a = 1 √ 1−v 2 for a soliton translating with a constant velocity v .The particle-like behavior of the solitons is a well-established fact but no consistent way to quantize the nonlinear waves and to extract the particle dynamics has been proposed.

Ivan Christov (NU)
Interacting sG solitons AIMS2008: SS #31 3 / 15 Some properties of the sine-Gordon solitons Faster solitons experience Lorentz contraction (a → ∞ as v → 1).Kinks are topological solitons.They are localized waves in the sense that Φ x is of finite energy (square-integrable).
N soliton solution satisfies Ivan Christov (NU) Interacting sG solitons AIMS2008: SS #31 4 / 15 The coarse-grain description Idea: "Degrade" the continuous description of the wave field to a discrete description involving "point" (quasi-)particles.
1 For a two-wave field we can formally write 2 Neglect Φ 12 because the interaction is accounted for by the X i s & a i s.
3 Compute L based on the resulting approximate expression for u.
4 Find the E-L equations for the collective coordinates which is almost the same as the one of two interacting point particles.

Discrete dynamics of the quasi-particles
After some simplification the equations of motion are ).The pseudomasses of the quasi-particles come about very naturally from the above framework to be Similarly, we can define the wave momentum and pseudomomentum: But wait, there's more: the crossmass of the two particles is defined as Interacting sG solitons AIMS2008: SS #31 6 / 15 Numerical results for two equal quasi-particles  Dynamics of a deforming quasi-particle If we do not ignore a(t), for a single soliton we have the following Lagrangian that governs the corresponding quasi-particle's dynamics E-L equations are: The solution space of these equations is quite rich, e.g., we recover Then, the 1 st equation gives d dt 8 √ 1−v 2 Ẋ = 0 (≡ Newton's 2 nd law for a relativistic particle!) Now the discrete Lagrangian is But, we cannot evaluate the integrals analytically as before.
Ivan Christov (NU) Interacting sG solitons AIMS2008: SS #31 9 / 15 Equations of motion of 2 QPs with 2 CVs each Highly nonlinear stiff system of equations that is not resolved with respect to the highest order derivatives.When using Mathematica's NDSolve numerical errors produce instabilities and the solutions blow up.
The physics of the soliton-soliton interaction (II) The solitons scatter, exchanging their phase speeds in the process.
But momentum has to be conserved ⇒ the QPs must exchange their pseudomasses (represented by the shape a i ).This is a 1D problem, so the QPs cannot scatter in an arbitrary direction (as in 2D or 3D).So, QP1(2) must absorb the energy in QP2(1)'s "internal mode" ⇒ exchange of the pseudomasses.So, the returning QP assumes the identity of the incoming QP; seems that the returning one is the incoming one passing through.
Featured example: the sine-Gordon equation Perring & Skyrme (1962) proposed the model unified field equation u tt − u xx = − sin u, which is the now-well-know sine-Gordon equation.It arises from a variational problem with the Lagrangian cos u − 1) dx.It has a one-soliton solution of the form (the kink) Φ(ξ; a) = 4 arctan[exp(aξ)], interaction; error ≈ 0.23%.