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1. Generalized matrix exponential solutions to the AKNS hierarchy

   Zhang, Jian-bing; Gu, Canyuan; Ma, Wen-Xiu (February 2018, Advances in mathematical physics)

   Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponential solutions share the same linear structure. 

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2. Diversity of interaction solutions to the (2+1)-dimensional Ito equation

   Ma, Wen-Xiu; Yong, Xuelin; Zhang, Hai-Qiang (January 2018, Computers & mathematics with applications)

   We aim to show the diversity of interaction solutions to the (2+1)-dimensional Ito equation, based on its Hirota bilinear form. The proof is given through Maple symbolic computations. An interesting characteristic in the resulting interaction solutions is the involvement of an arbitrary function. Special cases lead to lump solutions, lump-soliton solutions and lump-kink solutions. Two illustrative examples of the resulting solutions are displayed by three-dimensional plots and contour plots. 

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3. Soliton resonance and web structure in the Davey–Stewartson system

   https://doi.org/10.1088/1751-8121/ac78db  

   Biondini, Gino; Kireyev, Dmitri; Maruno, Ken-ichi (July 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract We write down and characterize a large class of nonsingular multi-soliton solutions of the defocusing Davey–Stewartson II equation. In particular we study their asymptotics at space infinities as well as their interaction patterns in the xy -plane, and we identify several subclasses of solutions. Many of these solutions describe phenomena of soliton resonance and web structure. We identify a subclass of solutions that is the analogue of the soliton solutions of the Kadomtsev–Petviashvili II equation. In addition to this subclass, however, we show that more general solutions exist, describing phenomena that have no counterpart in the Kadomtsev–Petviashvili equation, including V-shape solutions and soliton reconnection. 

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4. Holographic 6d co-dimension 2 defect solutions in M-theory

   https://doi.org/10.1007/JHEP11(2023)191  

   Gutperle, Michael; Klein, Nicholas; Rathore, Dikshant (November 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> We consider the uplift of co-dimension two defect solutions of seven dimensional gauged supergravity to eleven dimensions, previously found by two of the authors. The uplifted solutions are expressed as Lin-Lunin-Maldacena solutions and an infinite family of regular solutions describing holographic defects is found using the electrostatic formulation of LLM solutions. 

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5. Existence Theory for a Time-Dependent Mean Field Games Model of Household Wealth

   https://doi.org/10.1007/s00245-019-09619-5  

   Ambrose, David M. (October 2019, Applied Mathematics & Optimization)

   We study a nonlinear system of partial differential equations arising in macroeconomics which utilizes a mean field approximation. This system together with the corresponding data, subject to two moment constraints, is a model for debt and wealth across a large number of similar households, and was introduced in a recent paper of Achdou et al. (Philos Trans R Soc Lond Ser A 372(2028):20130397, 2014). We introduce a relaxation of their problem, generalizing one of the moment constraints; any solution of the original model is a solution of this relaxed problem. We prove existence and uniqueness of strong solutions to the relaxed problem, under the assumption that the time horizon is small. Since these solutions are unique and since solutions of the original problem are also solutions of the relaxed problem, we conclude that if the original problem does have solutions, then such solutions must be the solutions we prove to exist. Furthermore, for some data and for sufficiently small time horizons, we are able to show that solutions of the relaxed problem are in fact not solutions of the original problem. In this way we demonstrate nonexistence of solutions for the original problem in certain cases. 

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