An exploration of kinks/anti-kinks and breathers in deformed sine-Gordon models

Solitons refer to isolated waves that can travel without dissipating their energy. Basically, their behavior is similar to that of a particle under some conditions. Solitons and integrable systems play a vital role in the study of nonlinear phenomena since often they appear in the description of some physical systems. Moreover, the soliton attributes are intimately related to the integrability of the relevant nonlinear models in which they arise. Recent studies have shown that certain modified defocusing and focusing non-linear Schrödinger models, with dark and bright soliton solutions, exhibit the new feature of an infinite tower of exactly conserved charges. Furthermore, an analytical approach on inelastic solitary wave interactions for a quartic gKdV equation has revealed the absence of a pure 2-soliton solution in a special regime. Presently, researchers have unearthed that parity property is a sufficient condition in order to have the sequence of the exactly conserved charges in the kink-kink, kink-antikink and breather systems of deformed sine-Gordon models. As of now, based on the distinguishing new features associated to the non-relativistic deformed defocusing (focusing) non-linear Schrödinger with dark (bright) soliton solutions, as compared to the previous relativistic quasi-integrable models, there is need to advance studies in relation to this subject matter.

Professors Harold Blas (Institute of Physics) and Hector Flores Callisaya (Mathematics Department) from the Federal University of Mato Grosso in Cuiaba-Brazil, proposed a study whose main objective was to advance previous work in the field by utilizing a deformed sine-Gordon model. In addition, they focused on studying the space-reflection symmetries of some soliton solutions of deformed sine-Gordon models in the context of the quasi-integrability concept. Their work is currently published in the research journal, Communications in Nonlinear Science and Numerical Simulation.

The researchers commenced their work by introducing the relationship between the space-time parity and asymptotically conserved charges. Next, they clarified on the space-reflection parity symmetry and an order two automorphism of the sl(2) loop algebra relating the both dual sets of quasi-conserved quantities. They then proceeded to construct a tower of new exactly conserved charges for each field configuration possessing a definite space-reflection parity. Lastly, by considering linear combinations of the asymptotically conserved charges they showed, through analytical and numerical methods, that one tower became exactly conserved while the other one remained asymptotically-conserved after the combination.

Harold Blas and Hector Flores Callisaya were able to observe that from the numerical simulations undertaken, for the two-solitons without definite parity under space-reflection symmetry (kink-kink and kink-antikink scatterings with unequal and opposite velocities), there existed asymptotically conserved charges only. Additionally, they also noted that in the center-of-mass reference frame of the two solitons the parity symmetries and their associated set of exactly conserved charges could be restored. Lastly, they recorded that the positive parity breather-like (kink-antikink bound state) solution exhibited a tower of exactly conserved charges and a subset of charges which were periodic in time.

In conclusion, the Blas-Flores Callisaya study presented an in-depth demonstration of the quasi-integrability in deformed sine-Gordon models and infinite towers of conserved charges. The two researchers mainly observed that half of the infinite set of quasi-conserved quantities of the deformed sine-Gordon model were in fact exactly conserved, provided that some two-soliton configurations were eigenstates (even or odd) of the space-reflection operator. Altogether, the space-time and internal symmetries involved in the quasi-integrability phenomenon deserve further investigation, since they have potential applications in many areas of non-linear sciences, but those presented here have opened ways for new investigations on the nature of the quasi-integrability phenomena.