Bright Lump Research Articles

Magnetization reversal phenomenon of higher-order lump and mixed interaction structures on periodic background in the (2+1)-dimensional Heisenberg ferromagnet spin equation

This paper mainly researches a useful model that can describe the rapid magnetization reversal process, namely the (2+1)-dimensional Heisenberg ferromagnet spin equation. Based on its Lax pair, we construct the iterative generalized (r,N−r)-fold Darboux transformation (DT), and obtain new position-controllable higher-order lump, periodic wave and rich mixed lump–breather interaction structures on periodic background. In particular, with the increase of N in the iterative generalized (r,N−r)-fold DT, we find that the structure of the lump and periodic wave can flip over with respect to the background. Moreover, we also find a bright lump structure with two peaks and two valleys, which does not flip with the increase of N. We study the magnetization trajectory and magnetization on the Bloch sphere, from which we discover that the magnetization is distributed in an incomplete unit sphere for the flipped case, while the magnetization is distributed in a complete unit sphere for the non-flipped case. These magnetization reversal phenomena are expected to have potential applications in understanding the rapid magnetization reversal process.

Modulational Instability and Location Controllable Lump Solutions with Mixed Interaction Phenomena for the (2+1)-Dimensional Myrzakulov-Lakshmanan-IV Equation

Under consideration is a (2+1)-dimensional spin nonisospectral Myrzakulov-Lakshmanan-IV (ML-IV) equation, which has close relation with the celebrated nonlinear Schrödinger equation. In the first place we study the modulational instability (MI) for this equation and deduce its formation mechanism for diverse localized waves from the plane wave background. Secondly, in view of the known nonisospectral Lax pair of this equation, we first successfully extend the generalized (n, N-n)-fold Darboux Transformation (DT) from (1+1)-dimensional equation to this (2+1)-dimensional equation. As an application of the resulting DT, we derive some location controllable dark and bright lump, periodic wave and mixed breather-lump solutions, it is shown that the position of these localized waves can be controlled by some special parameters, so that we can move them to the positions we want. Especially, the dynamical behaviors of these localized waves are illustrated graphically. These results may have potential value for the study of applied ferromagnetism and nano magnetism.

Lump, multi-wave, kinky breathers, interactional solutions and stability analysis for general (2 + 1)-rth dispersionless Dym equation

The Lump, multi-wave, breather, interactional solutions and stability analysis for the general r-th dispersionless Dym equation are obtained by some fruitful transformations. This approach based on an hypothesis that includes a general quadratic polynomial function with some appropriate parameters. Also for finding multi-wave, breathers and interaction phenomena we use different assumptions that includes trigonometric and exponential functions. Eventually, lump, multi-wave, bright lump, dark lump and breather wave profiles of the solutions are analyzed. These results are drawn out graphically by choosing suitable different values of parameters with detailed behavior of physical structure. At the end, we also check the stability of the governing model.

Interaction phenomena between lump and solitary wave of a generalized (3 + 1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles**Project supported by National Natural Science Foundation of China (Grant No 81960715) and Science and Technology Project of Education Department of Jiangxi Province(GJJ151079).

In this paper, a generalized (3 + 1)-dimensional variable-coefficient nonlinear-wave equation is studied in liquid with gas bubbles. Based on the Hirota’s bilinear form and symbolic computation, lump and interaction solutions between lump and solitary wave are obtained, which include a periodic-shape lump solution, a parabolic-shape lump solution, a cubic-shape lump solution, interaction solutions between lump and one solitary wave, and between lump and two solitary waves. The spatial structures called the bright lump wave and the bright-dark lump wave are discussed. Interaction behaviors of two bright-dark lump waves and a periodic-shape bright lump wave are also presented. Their interactions are shown in some 3D plots.

Characteristics of lump solutions to a (3 + 1)-dimensional variable-coefficient generalized shallow water wave equation in oceanography and atmospheric science

In this paper, a (3 + 1) -dimensional variable-coefficient generalized shallow water wave equation in oceanography and atmospheric science is investigated. Lump solutions are presented based on the Hirota's bilinear form and symbolic computation, and characteristics of lump solutions, such as the bright lump wave and the bright-dark lump wave, are demonstrated in some 3D graphs and contour plots.

Nonautonomous characteristics of lump solutions for a [formula omitted]-dimensional Korteweg–de Vries equation with variable coefficients

We study a (2+1)-dimensional Korteweg–de Vries (KdV) equation with variable coefficients. By virtue of Hirota method, we present three types of nonautonomous lump solutions including the bright, bright–dark and dark lump ones. By considering different types of dispersion coefficients, we investigate the characteristics of trajectories, velocities and displacements of nonautonomous bright lump wave, which are different from the case of its constant-coefficient counterpart. We finally demonstrate the periodic attraction and repulsion interaction between a lump wave and a soliton. Our results might provide some physical insights into the relevant fields in nonlinear science.

Dark breather waves, dark lump waves and lump wave–soliton interactions for a [formula omitted]-dimensional generalized Kadomtsev–Petviashvili equation in a fluid

Fluids are seen in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, astrophysics and biology. In this paper, we investigate a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation for the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid. Breather-wave, lump-wave and lump wave–soliton solutions are derived under certain conditions via the Hirota method. With h1h2−1<0, where h1 and h2 represent the coefficients of dispersion and nonlinearity, respectively, we obtain the dark breather wave and lump wave. We observe the effects of h1, h2, h4, h6 and h8 on the dark breather wave and lump wave, where h6 is the perturbed effect, h4 and h8 stand for the disturbed wave velocity effects corresponding to the y and z coordinates: h1 and h2 influence the amplitude of the dark breather wave; h1, h4 and h8 influence the distance between the adjacent valleys of the dark breather wave; h1, h4, h6 and h8 influence the location of the dark breather wave; h2, h4, h6 and h8 influence the amplitude of the dark lump wave; h1, h4 and h8 influence the width of the dark lump wave; h4, h6 and h8 influence the location of the dark lump wave. When h1h2−1>0, we present the fusion between a bright lump wave and one bright soliton as well as fission of one bright soliton. We also observe the fusion between a dark lump wave and one dark soliton as well as fission of one dark soliton with h1h2−1>0.

Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation

In this work, a non-isospectral and variable-coefficient Kadomtsev–Petviashvili equation is considered using Hirota’s bilinear form and a direct assumption with arbitrary functions. Analytical rational solutions in light of positive quadratic functions and lump solutions of the variable-coefficient Kadomtsev–Petviashvili equation are obtained. These lump solutions describe two types of characters by some three-dimensional graphs and contour plots, which contain bright lump wave and bright–dark lump wave. Meanwhile, periodic structure of the lump wave is also shown.

Lump solutions to the ( $$\mathbf 2+1 $$ 2 + 1 )-dimensional Sawada–Kotera equation

In this paper, via generalized bilinear forms, we consider the (\(2+1\))-dimensional bilinear p-Sawada–Kotera (SK) equation. We derive analytical rational solutions in terms of positive quadratic functions. Through applying the dependent transformation, we present a class of lump solutions of the (\(2+1\))-dimensional SK equation. Those rationally decaying solutions in all space directions exhibit two kinds of characters, i.e., bright lump wave (one peak and two valleys) and bright–dark lump wave (one peak and one valley). In addition, we also obtain three families of bright–dark lump wave solutions to the nonlinear p-SK equation for \(p=3\).