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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. 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. A study of lump-type and interaction solutions to a (3+1)-dimensional Jimbo–Miwa-like equation

   https://doi.org/https://doi.org/10.1016/j.camwa.2018.07.008  

   Batwa, Sumayah; Ma, Wen-Xiu (October 2018, Computers & mathematics with applications)

   Using generalized bilinear equations, we construct a (3+1)-dimensional Jimbo–Miwa-like equation which possesses the same bilinear type as the standard (3+1)-dimensional Jimbo–Miwa equation. Classes of lump-type solutions and interaction solutions between lump-type and kink solutions to the resulting Jimbo–Miwa-like equation are generated through Maple symbolic computation. We discuss the conditions guaranteeing analyticity and positiveness of the solutions. By taking special choices of the involved parameters, 3D plots are presented to illustrate the dynamical features of the solutions. 

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5. Analytical construction of soliton families in one‐ and two‐dimensional nonlinear Schrödinger equations with nonparity‐time‐symmetric complex potentials

   https://doi.org/10.1111/sapm.12383  

   Yang, Jianke (April 2021, Studies in Applied Mathematics)

   Abstract The existence of soliton families in nonparity‐time‐symmetric complex potentials remains poorly understood, especially in two spatial dimensions. In this article, we analytically investigate the bifurcation of soliton families from linear modes in one‐ and two‐dimensional nonlinear Schrödinger equations with localized Wadati‐type nonparity‐time‐symmetric complex potentials. By utilizing the conservation law of the underlying non‐Hamiltonian wave system, we convert the complex soliton equation into a new real system. For this new real system, we perturbatively construct a continuous family of low‐amplitude solitons bifurcating from a linear eigenmode to all orders of the small soliton amplitude. Hence, the emergence of soliton families in these nonparity‐time‐symmetric complex potentials is analytically explained. We also compare these analytically constructed soliton solutions with high‐accuracy numerical solutions in both one and two dimensions, and the asymptotic accuracy of these perturbation solutions is confirmed. 

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