Superregular solitonic solutions of the defocusing nonlocal nonlinear Schrodinger equation and their novel perspectives

Duan-Yun Xia; Xiu-Bin Wang

Home/Archive/arXiv

arXiv · 2610.33560

Superregular solitonic solutions of the defocusing nonlocal nonlinear Schrodinger equation and their novel perspectives

Duan-Yun Xia·Xiu-Bin Wang

View original source ↗Read source document ↗Export metadata ↓

Abstract

Previous studies have indicated that the local defocusing nonlinear Schrödinger equation (NLSE) is modulationally stable. Therefore, superregular solitonic solutions characterizing the nonlinear evolution of modulational instability under weak localized perturbations do not exist. In this work, we study the nonlinear stage of the modulation instability of a condensate in the framework of the defocusing nonlocal NLSE. By using the dressing method and Joukowski transformation, we systematically derive the general $N$-solitonic solution of the defocusing nonlocal NLSE, and the explicit expressions for one-solitonic, two-solitonic and superregular two-solitonic solutions are provided. Then dynamics of these solutions is also examined. We find that these solutions show a much wider variety than those in the local NLSE, such as the one solitonic solution composed of two interacting wave components and the two-solitonic solution containing four wave components. Finally, the accuracy and noise robustness of the analytical solutions are confirmed by numerical simulations. Such results advance the theory of superregular solitonic solutions in nonlocal systems and offer new physical understanding of wave propagation in nonlocal nonlinear media.

Explore related subjects

nlin.PS math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Duan-Yun Xia, Xiu-Bin Wang. 2026-10-24. Superregular solitonic solutions of the defocusing nonlocal nonlinear Schrodinger equation and their novel perspectives. https://arxiv.org/abs/2610.33560

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Soliton gas resolution and statistics of random wave fields in semiclassical integrable turbulence

We develop a general analytical framework for determining the probability distribution of random nonlinear wave fields governed by the focusing nonlinear Schrödinger equation (fNLSE) in regimes where typical realizations are dominated by solitons. We formulate the soliton gas resolution conjecture for the long-time evolution of slowly varying ("semiclassical") random initial states and implement a stochastic analogue of the inverse scattering transform by establishing a relationship between the spectral density of states of the underlying bound state soliton gas and the probability density function (PDF) of the intensity of the resulting turbulent wave field. The derived explicit integral representation for the PDF is shown to be in excellent agreement with direct numerical simulations across several representative examples of fNLSE integrable turbulence. The analytical results of the paper are exact; they do not involve any closure assumptions, in contrast to classical turbulence theories. The results have broad applicability to physical systems including water waves, nonlinear optics, and superfluids, where integrable equations serve as important asymptotic models.

nlin.PS↗

arXiv2026

Power-Logistic Fronts in a Lossy Nonlinear Transmission Line

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 in the transmission line where the front propagates and the squared local sound speed. The latter plays an important role in our analysis, determining the admissibility conditions for the front. We specifically consider fronts with the power-logistic profile, including its hyperbolic-tangent special case. All physically admissible fronts of this form are shocks rather than kinks. For such shocks, the voltage--charge relation and the squared local sound speed--charge relation are expressed in terms of the lower incomplete beta function. We formulate the modified inverse-design method to calculate 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

Constant force amplifies pulse solitons via polarization coupling in mechanical metamaterials

Unlike kink solitons with nonzero topological charge, pulse solitons cannot be driven by the experimentally most easily implemented constant force because of their zero topological charge. Here, we focus on a vector soliton with two polarization components: a pulse soliton component (rotation) and a kink soliton component (translation). We apply a constant force at the mass centers of the units, thereby driving the translational component with a constant force. Numerical simulations show that energy injected into the translational component can be transferred simultaneously to the rotational component through coupling, thereby achieving equal amplification of positive and negative pulses. Next, we change the application of this constant force to eccentric loading and find that the initially symmetric amplification of positive and negative solitons gradually becomes asymmetric over time. These phenomena are all explained by a theoretical model based on the adiabatic approximation, which gives a quantitative relation for the amplitude evolution, while the higher-order correction terms arising from the coupling effect also explain the origin of asymmetric transmission under eccentric driving. Our results demonstrate that vector solitons possess a unique capability for pulse soliton amplification by a constant force through polarization coupling and that this coupling can further modulate the symmetry between positive and negative pulses.

nlin.PS↗