Collective coordinates in one-dimensional soliton models revisited

Abstract

We compare numerical solutions to the full field equations to simplified approaches based on implementing three collective coordinates for kink-antikink interactions within the ϕ 4 and φ 6 models in one time and one space dimensions. We particularly pursue the question whether the collective coordinate approximation substantiates the conjecture that vibrational modes are important for resonance structures to occur in kink-antikink scattering.

Multi-kink collisions in the ϕ 6 model

“…The (1+1)-dimensional φ 6 model is well-known in the literature [26,27,41,42,52], but the study of simultaneous multi-kink collisions in this model is carried out for the first time. A simultaneous collision of several kinks in a small region can produce a very large energy density in this region.…”

“…The interactions of kinks (and antikinks) are studied using different methods, in particular, quasi-exact methods such as the numerical solution of the equations of motion, which are partial differential equations, and approximate methods such as the collective coordinate approximation [27,[41][42][43][44][45][46] and the Manton's method [47][48][49]. The simple collective…”

Resonant interaction of ϕ4 kink with PT-symmetric perturbation with spatially periodic gain/loss coefficient

“…9 we plot the maximal amplitude of the kink's IM as a function of the kink initial velocity for different values of β from 2.5 to 6.0 with the step of 0. of Eqs. (15) and (16). The vertical lines show the resonance velocities found from Eq.…”

“…Similar effects are also possible in the φ 6 model in the absence of the kink's IM due to the existence of bound states in the spectrum of small oscillations about a combined kink-antikink configuration [15]; interestingly, such features were also present in the case of the parametrically modified sine-Gordon equation despite the potential absence of IMs in that case as well [4]. In the context of IM considerations, however, it is also timely and relevant to mention that recent works have disputed the validity and quantitative accuracy of such collective coordinate models [16]. When more than two Klein-Gordon kinks collide, the multibounce resonances and fractal soliton scattering are possible due to the radiationless energy exchange between the kinks in nearseparatrix collisions [17][18][19].…”

“…The kink velocity decreases when the IM amplitude increases, so that the kink translational and oscillation modes are coupled, as expected and as discussed previously. In fact, the collective coordinate equations (15)- (16) in some sense render this coupling explicit via their terms involving bothẊ (and φ K ) and a (and S).…”