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 transformations u = 2(ln f ) x and u = 2(ln f ) xx , where x is one spatial variable. Applications are made for a few generalized KP and BKP equations.where P is a polynomial of M variables and D = (D 1 , D 2 , · · · , D M ). Since the terms of odd powers are all zeros, we assume that P is an even polynomial, i.e., P (−x) = P (x), and to generate non-zero polynomial solutions, we require that P has no constant term, i.e., P (0) = 0. Moreover, we set

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A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the expfunction method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo-Miwa equation is treated, together with a Bäcklund transformation. MSC: 35Q58, 37K10, 35Q53

Some explicit traveling wave solutions to a Kolmogorov-PetrovskiiPiskunov equation are presented through two ansätze. By a Cole-Hopf transformation, this Kolmogorov-Petrovskii-Piskunov equation is also written as a bilinear equation and further two solutions to describe nonlinear interaction of traveling waves are generated. Bäcklund transformations of the linear form and some special cases are considered.

A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure which generalizes Hirota's perturbation scheme. With help of Maple, an application of the approach to the 3 + 1 dimensional potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of 2-wave and 3-wave solutions.