Pattern Formation and Solitons
- [1] arXiv:2405.19344 [pdf, ps, html, other]
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Title: On the modelling of short and intermediate water wavesComments: 8 pages, 1 figureJournal-ref: Applied Mathematics Letters 142 (2023) 108653Subjects: Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph)
The propagation of water waves of finite depth and flat bottom is studied in the case when the depth is not small in comparison to the wavelength. This propagation regime is complementary to the long-wave regime described by the famous KdV equation. The Hamiltonian approach is employed in the derivation of a model equation in evolutionary form, which is both nonlinear and nonlocal, and most likely not integrable. Possible implications for the numerical solutions are discussed.
- [2] arXiv:2405.19365 [pdf, ps, html, other]
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Title: Hydrodynamic modulation instability triggered by a two-wave systemSubjects: Pattern Formation and Solitons (nlin.PS)
The modulation instability (MI) is responsible for the disintegration of a regular nonlinear wave train and can lead to strong localizations in a from of rogue waves. This mechanism has been studied in a variety of nonlinear dispersive media, such as hydrodynamics, optics, plasma, mechanical systems, electric transmission lines, and Bose-Einstein condensates, while its impact on applied sciences is steadily growing. Following the linear stability analysis of weakly nonlinear waves, the classical MI dynamics, can be triggered when a pair of small-amplitude sidebands are excited within a particular frequency range around the main peak frequency. That is, a three-wave system is usually required to initiate the wave focusing process. Breather solutions of the nonlinear Schrödinger equation (NLSE) revealed that MI can generate much more complex localized structures, beyond the three-wave system initialization approach or by means of a continuous spectrum. In this work, we report an experimental study for deep-water surface gravity waves asserting that a MI process can be triggered by a single unstable sideband only, and thus, from a two-wave process when including the contribution of the peak frequency. The experimental data are validated against fully nonlinear hydrodynamic numerical wave tank simulations and show very good agreement. The long-term evolution of such unstable wave trains shows a distinct shift in the recurrent Fermi-Pasta-Ulam-Tsingou focusing cycles, which are captured by the NLSE and fully nonlinear hydrodynamic simulations with minor distinctions.
- [3] arXiv:2405.19370 [pdf, ps, html, other]
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Title: Interaction and adiabatic evolution of orthodromic and antidromic impulses in the axoplasmic fluidSubjects: Pattern Formation and Solitons (nlin.PS)
Unlike expected from the Hodgkin-Huxley model predictions, in which there is annihilation once orthodromic and antidromic impulses collide, the Heimburg-Jackson model demonstrates that both impulses penetrate each other as it has been shown experimentally. These impulses can be depicted as low amplitude nonlinear excitations in a weakly dissipative soliton model described by the damped NLSE. In view of the above, the Karpman-Solov'ev-Maslov perturbation theory turns out to be ideal to study the interaction and adiabatic evolution of orthodromic and antidromic impulses once axoplasmic fluid is present.
- [4] arXiv:2405.19478 [pdf, ps, html, other]
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Title: New sector morphologies emerge from anisotropic colony growthComments: 11 pages, 7 figuresSubjects: Pattern Formation and Solitons (nlin.PS); Biological Physics (physics.bio-ph); Populations and Evolution (q-bio.PE)
Competition during range expansions is of great interest from both practical and theoretical view points. Experimentally, range expansions are often studied in homogeneous Petri dishes, which lack spatial anisotropy that might be present in realistic populations. Here, we analyze a model of anisotropic growth, based on coupled Kardar-Parisi-Zhang and Fisher-Kolmogorov-Petrovsky-Piskunov equations that describe surface growth and lateral competition. Compared to a previous study of isotropic growth, anisotropy relaxes a constraint between parameters of the model. We completely characterize spatial patterns and invasion velocities in this generalized model. In particular, we find that strong anisotropy results in a distinct morphology of spatial invasion with a kink in the displaced strain ahead of the boundary between the strains. This morphology of the out-competed strain is similar to a shock wave and serves as a signature of anisotropic growth.
- [5] arXiv:2405.19887 [pdf, ps, html, other]
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Title: Modulational Instability of the Coupled Nonlinear volatility and option price modelC. Gaafele, Edmond B. Madimabe, K. Ndebele, P. Otlaadisa, B. Mozola, T. Matabana, K. Seamolo, P. PilaneSubjects: Pattern Formation and Solitons (nlin.PS)
We study the Coupled Nonlinear volatility and option price model via both Modulational instability analysis and direct simulations. Since the coupling term for both the volatility and the option price equation is the same, the MI results are dependent on it, and the stability of the volatility exists for the same condition as that of the price. The numerical simulations are done to comfirm the conditions of MI
- [6] arXiv:2405.20106 [pdf, ps, html, other]
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Title: Stability and dynamics of nonlinear excitations in a two-dimensional droplet-bearing environmentComments: 14 pages, 6 figuresSubjects: Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas)
We unravel stationary states in the form of dark soliton stripes, bubbles, and kinks embedded in a two-dimensional droplet-bearing setting emulated by an extended Gross-Pitaevskii approach. The existence of these configurations is corroborated through an effectively reduced potential picture demonstrating their concrete parametric regions of existence. The excitation spectra of such configurations are analyzed within the Bogoliubov-de-Gennes framework exposing the destabilization of dark soliton stripes and bubbles, while confirming the stability of droplets, and importantly unveiling spectral stability of the kink against transverse excitations. Additionally, a variational approach is constructed providing access to the transverse stability analysis of the dark soliton stripe for arbitrary chemical potentials and widths of the structure. This is subsequently compared with the stability analysis outcome demonstrating very good agreement at small wavenumbers. Dynamical destabilization of dark soliton stripes via the snake instability is showcased, while bubbles are found to feature both a splitting into a gray soliton pair and a transverse instability thereof. These results shed light on unexplored stability and instability properties of nonlinear excitations in environments featuring a competition of mean-field repulsion and beyond-mean-field attraction that can be probed by state-of-the-art experiments.
New submissions for Friday, 31 May 2024 (showing 6 of 6 entries )
- [7] arXiv:2405.19607 (cross-list from cond-mat.quant-gas) [pdf, ps, html, other]
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Title: Generic transverse stability of kink structures in atomic and optical nonlinear media with competing attractive and repulsive interactionsComments: 4 pages, 3 figuresSubjects: Quantum Gases (cond-mat.quant-gas); Pattern Formation and Solitons (nlin.PS); Atomic Physics (physics.atom-ph)
We demonstrate the existence and stability of one-dimensional (1D) topological kink configurations immersed in higher-dimensional bosonic gases and nonlinear optical setups. Our analysis pertains, in particular, to the two- and three-dimensional extended Gross-Pitaevskii models with quantum fluctuations describing droplet-bearing environments but also to the two-dimensional cubic-quintic nonlinear Schrödinger equation containing higher-order corrections to the nonlinear refractive index. Contrary to the generic dark soliton transverse instability, the kink structures are generically robust under the interplay of low-amplitude attractive and high-amplitude repulsive interactions. A quasi-1D effective potential picture dictates the existence of these defects, while their stability is obtained through linearization analysis and direct dynamics in the presence of external fluctuations showcasing their unprecedented resilience. These generic (across different models) findings should be detectable in current cold atom and optics experiments. They also offer insights towards controlling topological excitations and their usage in topological quantum computers.
- [8] arXiv:2405.20265 (cross-list from math.AP) [pdf, ps, html, other]
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Title: Pinning and dipole asymptotics of locally deformed striped phasesComments: 30pSubjects: Analysis of PDEs (math.AP); Pattern Formation and Solitons (nlin.PS)
We investigate the effect of spatial inhomogeneity on perfectly periodic, self-organized striped patterns in spatially extended systems. We demonstrate that inhomogeneities select a specific translate of the striped patterns and induce algebraically decaying, dipole-type farfield deformations. Phase shifts and leading order terms are determined by effective moments of the spatial inhomogeneity. Farfield decay is proportional to the derivatives of the Green's function of an effective Laplacian. Technically, we use mode filters and conjugacies to an effective Laplacian to establish Fredholm properties of the linearization in Kondratiev spaces. Spatial localization in a contraction argument is gained through the use of an explicit deformation ansatz and a subtle cancellation in Bloch wave space.
Cross submissions for Friday, 31 May 2024 (showing 2 of 2 entries )
- [9] arXiv:2404.16715 (replaced) [pdf, ps, html, other]
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Title: Third order interactions shift the critical coupling in multidimensional Kuramoto modelsComments: 15 pages, 4 figures. New analysis of bi-stability and hysteresisSubjects: Pattern Formation and Solitons (nlin.PS)
The study of higher order interactions in the dynamics of Kuramoto oscillators has been a topic of intense recent research. Arguments based on dimensional reduction using the Ott-Antonsen ansatz show that such interactions usually facilitate synchronization, giving rise to bi-stability and hysteresis. Here we show that three body interactions shift the critical coupling for synchronization towards higher values in all dimensions, except $D=2$, where a cancellation occurs. After the transition, three and four body interactions combine to facilitate synchronization. Similar to the 2-dimensional case, bi-stability and hysteresis develop for large enough higher order interactions. We show simulations in $D=3$ and $4$ to illustrate the dynamics.