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This paper aims to present two hierarchies of nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. The key point is to make similarity transformations for the spectral matrix, which keep the associated zero curvature equations invariant and then there follow reduced nonlocal integrable mKdV-type equations. The success lies in determining a Laurent series solution to the corresponding reduced stationary zero curvature equation, which generates temporal matrix spectral problems in the zero curvature formation. [ABSTRACT FROM AUTHOR]

Starting from a kind of higher-order matrix spectral problems, we generate integrable Hamiltonian hierarchies through the zero-curvature formulation. To guarantee the Liouville integrability of the obtained hierarchies, the trace identity is used to establish their Hamiltonian structures. Illuminating examples of coupled nonlinear Schrödinger equations and coupled modified Korteweg–de Vries equations are worked out. [ABSTRACT FROM AUTHOR]

A new generalized Kadomtsev–Petviashvili (GKP) equation is derived from a bilinear differential equation by taking the transformation u = 2 (ln f) x . By symbolic computation with Maple, lump solutions, rationally localized in all directions in the space, to the GKP equation are presented. The obtained lump solutions contain a set of six free parameters, four of which should satisfy a nonzero determinant condition. As special examples, six particular lump solutions are constructed and depicted with t = 1. [ABSTRACT FROM AUTHOR]

We aim to construct exact and explicit solutions to a generalized Bogoyavlensky-Konopelchenko equation through the Maple computer algebra system. The considered nonlinear equation is transformed into a Hirota bilinear form, and symbolic computations are made for solving both the nonlinear equation and the corresponding bilinear equation. A few classes of exact and explicit solutions are generated from different ansätze on solution forms, including traveling wave solutions, two-wave solutions, and polynomial solutions. [ABSTRACT FROM AUTHOR]

A bilinear form of a nonisospectral differential-difference equation related to the Ablowitz-Ladik (AL) spectral problem is derived by a transformation of dependent variables. Exact solutions to the resulting bilinear equation are found. The -soliton-like solutions and the double Casoratian solutions are derived by means of Hirota's direct method and the double Casoratian technique, respectively. Moreover, the connection between those two classes of solutions is explored. [ABSTRACT FROM AUTHOR]

By using the Casoratian technique, we construct the double Casoratian solutions whose entries satisfy matrix equation of a differential-difference equation related to the Ablowitz-Ladik spectral problem. Soliton solutions and rational-like solutions are obtained from taking special cases in general solutions. [ABSTRACT FROM AUTHOR]