Solitons, which are solutions of the equations of motion of integrable systems, do not interact, that's their defining property. The construction of the N-soliton state in terms of 1-soliton states is not at all trivial, precisely because, since the equations of motion are non-linear, the superposition principle does not apply. One example is here https://arxiv.org/abs/2411.15487

See also here https://www.researchgate.net/publication/362930117_HIROTA_METHOD_AND_SOLITON_SOLUTIONS

Solitons are the outcome of non-linear dispersive systems; they usually are constant, localized wave packets that live stable and keep their uniqueness through propelling as well as interaction. Understanding the progression of solitons necessitates the comprehension of the concepts of integrability as articulated by meriting capabilities of analytic solutions and offers profound insights into nonlinear phenomena. Integrable systems allow for accurate solutions according to their intrinsic infinite conserved qualities alongside the nonlinear Schrödinger equation (NLSE), the Korteweg–de Vries (KdV) equation, and the sine-Gordon equation.

These represent a set of non-linear partial differential equations that accurately model soliton behaviors in all dimensions (Ablowitz & Segur, 1981). For instance, they model wave movement in nonlinear media like shallow water, optical fibers, as well as case at hand, which is blood flow oscillations in large vessels. With respect to the sine-Gordon system (SG) and the nonlinear Schröinger equation (NLS), we noted that relatively short-span waves exist, which were relatively efficient in transporting momentum and energy, and suddenly convert to solitons.

According to the sine-Gordon (SG) system and the NLS, solitons can be decomposed into dispersed wave excitations (wave radiation). The wave-like relics from combustion could occasionally be likened to solitons as they propagate pretty far away from the combustion origin. In relation to the plasma wave, solitons are central because they delineate the collapse phenomenon into regular fine structures (Ben Lkehal et al., 2020). These solitons would conveniently rearrange with size, stretching as well as merging over the whole reactant region. A rational coefficient of the sine-Gordon equation was also identified in the model. We would regularly change the coefficient to derive additional types of solitons and discover more on the vital role that the rational coefficient has on these models. Alternatively, the NLS delineates the elastic medium's motion.

References

Ablowitz, M. J., & Segur, H. (1981). Solitons and the Inverse Scattering Transform. SIAM. Chapter 8. Ben

Lkehal, M., Mohamed, I., & Belabid, D. (2020). A note on the solutions of the Zakharov–Kuznetsov equation. Nonlinear dynamics, 100(1), 217–220.