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1. Rogue wave patterns associated with Adler–Moser polynomials in the nonlinear Schrödinger equation

   https://doi.org/10.1016/j.aml.2023.108871  

   Yang, Bo; Yang, Jianke (February 2024, Applied Mathematics Letters)

   We report new rogue wave patterns in the nonlinear Schrödinger equation. These patterns include heart-shaped structures, fan-shaped sectors, and many others, that are formed by individual Peregrine waves. They appear when multiple internal parameters in the rogue wave solutions get large. Analytically, we show that these new patterns are described asymptotically by root structures of Adler–Moser polynomials through a dilation. Since Adler–Moser polynomials are generalizations of the Yablonskii–Vorob’ev polynomial hierarchy and contain free complex parameters, these new rogue patterns associated with Adler–Moser polynomials are much more diverse than previous rogue patterns associated with the Yablonskii–Vorob’ev polynomial hierarchy. We also compare analytical predictions of these patterns to true solutions and demonstrate good agreement between them. 

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2. Existence of solitary waves in particle lattices with power-law forces

   https://doi.org/10.1088/1361-6544/ad8c1c  

   Ingimarson, Benjamin; Pego, Robert_L (November 2024, Nonlinearity)

   Abstract We prove the existence of small solitary waves for one-dimensional lattices of particles that each repel every other particle with a force that decays as a power of distance. For force exponentsα + 1 with $\frac{4}{3}<\alpha <3$, we employ fixed-point arguments to find near-sonic solitary waves having scaled velocity profiles close to non-degenerate solitary-wave profiles of fractional KdV or generalized Benjamin–Ono equations. These equations were recently found to approximately govern unidirectional long-wave motions in these lattices. 

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3. Onset of energy equipartition among surface and body waves

   https://doi.org/doi.org/10.1098/rspa.2020.0775  

   Borcea, L; Garnier, J; Solna, K. (April 2021, Proceedings of the Royal Society of London)

   null (Ed.)

   We derive a radiative transfer equation that accounts for coupling from surface waves to body waves and the other way around. The model is the acoustic wave equation in a two-dimensional waveguide with reflecting boundary. The waveguide has a thin, weakly randomly heterogeneous layer near the top surface, and a thick homogeneous layer beneath it. There are two types of modes that propagate along the axis of the waveguide: those that are almost trapped in the thin layer, and thus model surface waves, and those that penetrate deep in the waveguide, and thus model body waves. The remaining modes are evanescent waves. We introduce a mathematical theory of mode coupling induced by scattering in the thin layer, and derive a radiative transfer equation which quantifies the mean mode power exchange.We study the solution of this equation in the asymptotic limit of infinite width of the waveguide. The main result is a quantification of the rate of convergence of the mean mode powers toward equipartition. 

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4. General rogue waves of infinite order: exact properties, asymptotic behaviour, and effective numerical computation

   https://doi.org/10.1017/jnw.2025.10018  

   Bilman, Deniz; Miller, Peter D (November 2025, Journal of Nonlinear Waves)

   This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in $$L^2(\mathbb{R})$$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper, we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately. 

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5. A rigorous mathematical theory for topological phases and edge modes in spring-mass mechanical systems

   https://doi.org/10.1098/rspa.2023.0910  

   Ozdemir, Ridvan; Lin, Junshan (April 2024, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   In this work, we examine the topological phases of the spring-mass lattices when the spatial inversion symmetry of the system is broken and prove the existence of edge modes when two lattices with different topological phases are glued together. In particular, for the one-dimensional lattice consisting of an infinite array of masses connected by springs, we show that the Zak phase of the lattice is quantized, only taking the value 0 or $\pi$. We also prove the existence of an edge mode when two semi-infinite lattices with distinct Zak phases are connected. For the two-dimensional honeycomb lattice, we characterize the valley Chern numbers of the lattice when the masses on the lattice vertices are uneven. The existence of edge modes is proved for a joint honeycomb lattice formed by gluing two semi-infinite lattices with opposite valley Chern numbers together. 

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