Vector semi-rational rogon-solitons and asymptotic analysis for any multi-component Hirota equations with mixed backgrounds

The Hirota equation can be used to describe the wave propagation of an ultrashort optical field. In this paper, the multi-component Hirota (alias n-Hirota, i.e. n-component third-order nonlinear Schrödinger) equations with mixed non-zero and zero boundary conditions are explored. We employ the multiple roots of the characteristic polynomial related to the Lax pair and modified Darboux transform to find vector semi-rational rogon-soliton solutions (i.e. nonlinear combinations of rogon and soliton solutions). The semi-rational rogon-soliton features can be modulated by the polynomial degree. For the larger solution parameters, the first m (m < n) components with non-zero backgrounds can be decomposed into rational rogons and grey-like solitons, and the last n − m components with zero backgrounds can approach bright-like solitons. Moreover, we analyze the accelerations and curvatures of the quasi-characteristic curves, as well as the variations of accelerations with the distances to judge the interaction intensities between rogons and grey-like solitons. We also find the semi-rational rogon-soliton solutions with ultra-high amplitudes. In particular, we can also deduce vector semi-rational solitons of the n-component complex mKdV equation. These results will be useful to further study the related nonlinear wave phenomena of multi-component physical models with mixed background, and even design the related physical experiments.     

High-order breather,M-kink lump and semi-rational solutions of potential Kadomtsev-Petviashvili equation

N-kink soliton and high-order synchronized breather solutions for potential Kadomtsev-Petviashvili equation are derived by means of the Hirota bilinear method,and the limit process of high-order synchronized breathers are shown.Furthermore,M-lump solutions are also presented by taking the long wave limit.Additionally,a family of semi-rational solutions with elastic collision are generated by taking a long-wave limit of only a part of exponential functions,their interaction behaviors are shown by three-dimensional plots and contour plots.     

General soliton and (semi-)rational solutions to the nonlocal Mel'nikov equation on the periodic background

In this paper, the Hirota's bilinear method and Kadomtsev-Petviashvili hierarchy reduction method are applied to construct soliton, line breather and (semi-)rational solutions to the nonlocal Mel'nikov equation with nonzero boundary conditions. These solutions are expressed as Gram-type determinants. When N is even, soliton, line breather and (semi-)rational solutions on the constant background are derived while these solutions are located on the periodic background for odd N. Regularity of these solutions and their connections with the local Mel'nikov equation are analyzed for proper choices of parameters that appear in the solutions. The dynamics of the solutions are discussed in detail. All possible configurations of soliton and lump solutions are found for . Several interesting dynamical behaviors of semi-rational solutions are observed. It is shown that certain lumps may exhibit fusion and fission phenomena during their interactions with solitons while some lump may change its direction of movement after it collides with solitons.     

Dynamical analysis of rational and semi-rational solution for a new extended (3 + 1)-dimensional Kadomtsev-Petviashvili equation

This paper proposes a new extended (3 + 1)-dimensional Kadomtsev-Petviashvili equation that portrays a unique dispersion effect about $x,z$. Its integrability is confirmed via the WTC-Kruskal algorithm in Painlevé sense. $N$-soliton, breather, and $O$-type solitary wave are derived systematically at first. Then, the mixed solution composed of soliton and breather is obtained. In addition, the “long wave” limit is employed to construct rational and semi-rational solution. The rational solution can be classified as rogue wave, $T$-type solitary wave, and lump wave. The semi-rational solution has the form a hybrid of two solitons, a hybrid of rogue wave and soliton, a hybrid of lump and soliton(s), and a hybrid of lump and breather. The results may help simulate complex waves and their interactions in fluid.