Study of lump and lump-kink solitons of a coupled reduced Hirota bilinear equation

By symbolic computation and searching for the solutions of the positive quadratic functions of the related bilinear equations, two kinds of lump solutions of the (3[Formula: see text]+[Formula: see text]1)-dimensional weakly coupled Hirota bilinear equation are derived, and the practicability of this method is verified. Then we add an exponential function to the original positive quadratic function, and obtain a new solution of the Hirota bilinear equation. The interaction between the lump solutions and lump-kink solutions is included in the new solution. On this basis, we give the possibility of adding multiple exponential functions. Finally, we give the coupled reduced Hirota bilinear equation lump-kink solitons by combining the above two methods. In order to ensure the analyticity and reasonable localization of the block, two sets of necessary and sufficient conditions are given for the parameters involved in the solution. The local characteristics and energy distribution of bulk solution are analyzed and explained.

Rational wave solutions to a generalized (2+1)-dimensional Hirota bilinear equation

A generalized form of (2+1)-dimensional Hirota bilinear (2D-HB) equation is considered herein in order to study nonlinear waves in fluids and oceans. The present goal is carried out through adopting the simplified Hirota’s method as well as ansatz approaches to retrieve a bunch of rational wave structures from multiple soliton solutions to breather, rational, and complexiton solutions. Some figures corresponding to a series of rational wave structures are provided, illustrating the dynamics of the obtained solutions. The results of the present paper help to reveal the existence of rational wave structures of different types for the 2D-HB equation.

A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

Abundant mixed lump-kink solutions to a (2+1)-dimensional bSK equation

In this paper, we study lump-kink solutions of a (2+1)-dimensional bidirectional Sawada–Kotera equation and discuss their dynamics. A Hirota bilinear form of a (2+1)-dimensional bidirectional Sawada–Kotera equation is deduced via a dependent logarithmic transformation. Based on this Hirota bilinear equation, we obtain eight classes of lump-kink solutions which combine stripe soliton and lump soliton by using symbolic computations. Our simulation results with the appropriate choice of the arbitrary parameters that show the motion of lump soliton and the process of interaction between lump soliton and a stripe soliton.