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1. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms

   Ma, Wen-Xiu; Zhou, Yuan (February 2018, Journal of differential equations)

   Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable logarithmic derivative transformations. Applications are made for a few generalized KP and BKP equations. 

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2. A local discontinuous Galerkin method for nonlinear parabolic SPDEs

   https://doi.org/10.1051/m2an/2020026  

   Li, Yunzhang; Shu, Chi-Wang; Tang, Shanjian (January 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L 2 -stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ( O ( h (k+1) )) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method. 

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3. Implicit Adaptive Mesh Refinement for Dispersive Tsunami Propagation

   https://doi.org/10.1137/23M1585210  

   Berger, Marsha J; LeVeque, Randall J (August 2024, SIAM Journal on Scientific Computing)

   We present an algorithm to solve the dispersive depth-averaged Serre--Green--Naghdi equations using patch-based adaptive mesh refinement. These equations require adding additional higher derivative terms to the nonlinear shallow water equations. This has been implemented as a new component of the open source GeoClaw software that is widely used for modeling tsunamis, storm surge, and related hazards, improving its accuracy on shorter wavelength phenomena. We use a formulation that requires solving an elliptic system of equations at each time step, making the method implicit. The adaptive algorithm allows different time steps on different refinement levels and solves the implicit equations level by level. Computational examples are presented to illustrate the stability and accuracy on a radially symmetric test case and two realistic tsunami modeling problems, including a hypothetical asteroid impact creating a short wavelength tsunami for which dispersive terms are necessary. 

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4. Unfinished business in a nonlinear sigma model on de Sitter background

   https://doi.org/10.1007/JHEP06(2023)206  

   Woodard, R. P.; Yesilyurt, B. (July 2023, Journal of High Energy Physics)

   A bstract Nonlinear sigma models on de Sitter background possess the same kind of derivative interactions as gravity, and show the same sorts of large spacetime logarithms in correlation functions and solutions to the effective field equations. It was recently demonstrated that these logarithms can be resummed by combining a variant of Starobinsky’s stochastic formalism with a variant of the renormalization group. This work considers one of these models and completes two pieces of analysis which were left unfinished: the evolution of the background at two loop order and the one loop beta function. 

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5. Existence results for fractional order functional differential equations with infinite delay in the sense of the deformable derivative

   Etefa, Mesfin; N'guérékata, Gaston M. (January 2022, Analele Universităţii din Oradea Fascicula Matematică)

   In this article, we discuss the existence and uniqueness of solutions for initial value problems of fractional order functional and neutral functional differential equations with infinite delay. We use the deformable derivative introduced in 2017 by Zulfeqarr et. al (see [21]). Our results are obtained using the Banach fixed point theorem and the nonlinear alternative Leray-Schauder type theorem. We provide an example as an illustration of the main results. 

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