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1. Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation

   Yang, Jin-Yun ; Ma, Wen-Xiu ; Qin, Zhenyun ( September 2018 , Analysis and mathematical physics)

   Based on the Hirota bilinear form of the (2+1)-dimensional Ito equation, one class of lump solutions and two classes of interaction solutions between lumps and line solitons are generated through analysis and symbolic computations with Maple. Analyticity is naturally guaranteed for the presented lump and interaction solutions, and the interaction solutions reduce to lumps (or line solitons) while the hyperbolic-cosine (or the quadratic function) disappears. Three-dimensional plots and contour plots are made for two specific examples of the resulting interaction solutions. 

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2. Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation

   Chen, Shou-Ting ; Ma, Wen-Xiu ( June 2018 , Frontiers of Mathematics in China)

   A (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation that possesses a Hirota bilinear form is considered. Starting with its Hirota bilinear form, a class of explicit lump solutions is computed through conducting symbolic computations with Maple, and a few plots of a specicpresented lump solution are made to shed light on the characteristics of lumps. The result provides a new example of (2 + 1)-dimensional nonlinear partial differential equations which possess lump solutions. 

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3. 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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4. Lump chains in the KP‐I equation

   https://doi.org/10.1111/sapm.12420  

   Lester, Charles ; Gelash, Andrey ; Zakharov, Dmitry ; Zakharov, Vladimir ( July 2021 , Studies in Applied Mathematics)

   We construct a broad class of solutions of the Kadomtsev–Petviashvili (KP)‐I equation by using a reduced version of the Grammian form of the‐function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP‐I analogues of the family of line‐soliton solutions of KP‐II. However, the linear arrangements that we construct for KP‐I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We also construct solutions describing interactions between lump chains and individual lumps, and discuss the relationship between the solutions obtained using the reduced and regular Grammian forms.

    

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5. Evolution of truncated and bent gravity wave solitons: the Mach expansion problem

   https://doi.org/10.1017/jfm.2020.952  

   Ryskamp, Samuel ; Maiden, Michelle D. ; Biondini, Gino ; Hoefer, Mark A. ( February 2021 , Journal of Fluid Mechanics)

   null (Ed.)

   The dynamics of initially truncated and bent line solitons for the Kadomtsev–Petviashvili (KPII) equation modelling internal and surface gravity waves is analysed using modulation theory. In contrast to previous studies on obliquely interacting solitons that develop from acute incidence angles, this work focuses on initial value problems for the obtuse incidence of two or three partial line solitons, which propagate away from one another. Despite counterpropagation, significant residual soliton interactions are observed with novel physical consequences. The initial value problem for a truncated line soliton – describing the emergence of a quasi-one-dimensional soliton from a wide channel – is shown to be related to the interaction of oblique solitons. Analytical descriptions for the development of weak and strong interactions are obtained in terms of interacting simple wave solutions of modulation equations for the local soliton amplitude and slope. In the weak interaction case, the long-time evolution of truncated and large obtuse angle solitons exhibits a decaying, parabolic wave profile with temporally increasing focal length that asymptotes to a cylindrical Korteweg–de Vries soliton. In contrast, the strong interaction case of slightly obtuse interacting solitons evolves into a steady, one-dimensional line soliton with amplitude reduced by an amount proportional to the incidence slope. This strong interaction is identified with the ‘Mach expansion’ of a soliton with an expansive corner, contrasting with the well-known Mach reflection of a soliton with a compressive corner. Interestingly, the critical angles for Mach expansion and reflection are the same. Numerical simulations of the KPII equation quantitatively support the analytical findings. 

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