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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. 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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3. 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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4. The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity

   https://doi.org/10.15781/chkn-sn69  

   Ifraim, Mihaela; Rowan, James; Tataru, Daniel; Wan, Lizhe (January 2022, Ars Inveniendi Analytica)

   This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis. 

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5. Non-Hermitian skin effect in two dimensional continuous systems

   https://doi.org/10.1088/1402-4896/aca43b  

   Yuce, C; Ramezani, H (December 2022, Physica Scripta)

   Abstract An extensive number of the eigenstates can become exponentially localized at one boundary of nonreciprocal non-Hermitian systems. This effect is known as the non-Hermitian skin effect and has been studied mostly in tight-binding lattices. To extend the skin effect to continues systems beyond 1D, we introduce a quadratic imaginary vector potential in the continuous two dimensional Schrödinger equation. We find that inseparable eigenfunctions for separable nonreciprocal Hamiltonians appear under infinite boundary conditions. Introducing boundaries destroy them and hence they can only be used as quasi-stationary states in practice. We show that all eigenstates can be clustered at the point where the imaginary vector potential is minimum in a confined system. 

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