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1. Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity

   https://doi.org/10.1088/1361-6544/ac93e1  

   Friedman, Isaac; Riaño, Oscar; Roudenko, Svetlana; Son, Diana; Yang, Kai (December 2022, Nonlinearity)

   Abstract We consider two types of the generalized Korteweg–de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of H 1 that includes functions with polynomial decay, extending the result of Linares et al (2019 Commun. Contemp. Math. 21 1850056) to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. We include a comparison of solutions to both types of equations, in particular, we investigate soliton resolution for the positive and negative data with different decay rates. Finally, we study the interaction of various solitary waves in both models, showing the formation of solitons, dispersive radiation and even breathers, all of which are easier to track in nonlinearities with lower power. 

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2. Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones

   Conlon, Ronan J.; Deruelle, Alix (June 2020, ArXivorg)

   We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient Kähler-Ricci soliton in every Kähler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient Kähler-Ricci soliton on the cone. 

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3. Directly accessing the single-soliton state of a Kerr microcomb and its universal scaling law [Invited]

   https://doi.org/10.1364/OME.541224  

   Sun, Wenhan; Li, Jingwei; Wang, Ruixuan; Li, Qing (November 2024, Optical Materials Express)

   Soliton microcombs have attracted considerable research interest due to their unique properties. Being able to directly access the single-soliton state in a Kerr microresonator simplifies the device operation and may inspire new applications. However, the general conditions leading to such operations are not well understood. In this work, we aim to elucidate the key factors enabling the direct access of the single-soliton state in a Kerr microresonator by combining the experimental results in an integrated silicon carbide platform and a comprehensive analysis based on the normalized Lugiato-Lefever equation. A general criterion linking the Kerr nonlinearity, dispersion, and thermo-optic properties has been derived, which is applicable to Kerr microresonators with varied materials, sizes, optical quality factors, and dispersion. 

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4. Weakly nonlinear topological gap solitons in Su–Schrieffer–Heeger photonic lattices

   https://doi.org/10.1364/OL.411102  

   Guo, Min; Xia, Shiqi; Wang, Nan; Song, Daohong; Chen, Zhigang; Yang, Jianke (November 2020, Optics Letters)

   We study both theoretically and experimentally the effect of nonlinearity on topologically protected linear interface modes in a photonic Su–Schrieffer–Heeger (SSH) lattice. It is shown that under either focusing or defocusing nonlinearity, this linear topological mode of the SSH lattice turns into a family of topological gap solitons. These solitons are stable. However, they exhibit only a low amplitude and power and are thus weakly nonlinear, even when the bandgap of the SSH lattice is wide. As a consequence, if the initial beam has modest or high power, it will either delocalize, or evolve into a soliton not belonging to the family of topological gap solitons. These theoretical predictions are observed in our experiments with optically induced SSH-type photorefractive lattices. 

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5. Existence and higher regularity of statistically steady states for the stochastic Coleman-Gurtin equation

   https://doi.org/10.3934/eect.2025038  

   Glatt-Holtz, Nathan E; Martinez, Vincent R; Nguyen, Hung D (January 2025, Evolution Equations and Control Theory)

   We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way. 

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