Tip-splitting evolution in the idealized Saffman-Taylor problem

We investigate the interaction of dissipative Kerr solitons in two nearly identical coherently driven, coupled microresonators in the presence of third-order dispersion (TOD). Using an asymptotic reduction of the coupled Lugiato--Lefever equations, we derive effective interaction equations governing the soliton separation and collective drift for weak intercavity coupling. TOD breaks the parity symmetry of the soliton profile and qualitatively modifies the effective interaction, leading to asymmetric interaction forces and multiple stable and unstable soliton clusters. The existence and stability of these clusters are controlled by the relative phase of the driving fields and the group-velocity mismatch between the resonators. In particular, an appropriate value of the group-velocity mismatch can give rise to multiple stable bound states, whereas deviations from this value trigger successive saddle-node bifurcations, ultimately leading to periodic soliton collisions. The analytical predictions are confirmed by numerical simulations of the full coupled system.

nlin.PS↗

arXiv2026

Shocks in a Lossy Nonlinear Transmission Line: The Inverse-Design Approach

We consider a lossy transmission line with a nonlinear voltage--charge relation. In the framework of the inverse-design method, starting from a prescribed monotonic front profile, we determine exactly the voltage--charge relation and the squared local sound speed in the transmission line where the front propagates. The latter plays an important role in our analysis, determining the admissibility conditions for the front. We specifically consider fronts with the profiles belonging to the four-parameter asymmetric power-logistic family. For such fronts the voltage--charge relation and the squared local sound speed can be expressed in terms of the lower incomplete beta function. All physically admissible fronts of this form are shocks rather than kinks. We formulate an inverse-design-motivated method for calculating the profile of a shock propagating in the transmission line with a prescribed voltage--charge relation. We examine the local spectral properties of the front propagating in the transmission line in terms of the original partial differential equations and show that the short-wavelength perturbation modes are damped. The kink--shock distinction obtained from the mechanical analogy is independently confirmed through the long-wavelength characteristic analysis.

nlin.PS↗

arXiv2026

Asymptotic breathers in the Manakov system

We construct vector (two-component) asymptotic Akhmediev breathers in the Manakov system, matching generalized modulational-instability (gMI) eigenmodes to decaying tails of the breather states. The branch-resolved matching provides analytical results linking the gMI-perturbation envelopes to the positions of the emerging breathers. Numerical simulations of the system with the focusing and defocusing nonlinearities corroborate the analytical predictions for each MI branch. We further develop the asymptotic theory for two-breather solutions generated by two independent gMI modes, linking their temporal separation to the input envelopes and interaction-induced shifts. Further numerical simulations reveal that breather patterns exhibit robustness in the presence of weak nonintegrable perturbations. These findings establish a framework relating experimentally controllable input modulations to the branch-selective formation and positioning of breathers in multicomponent nonlinear systems.

nlin.PS↗