Mixed 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.

Bilinear auto-Bäcklund transformations, breather and mixed lump–kink solutions for a (3+1)-dimensional integrable fourth-order nonlinear equation in a fluid

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Bäcklund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump–kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids.

Mixed lump–kink solutions and lump–soliton solutions to the generalized BKP equation with some second order terms

A generalized ( 2 + 1 ) -dimensional BKP equation with more second order nonlinear terms is studied. The different types of lump–kink solutions and lump–soliton solutions are constructed by using a combination of Hirota’s bilinear method and Maple symbolic computations. By assuming the solution of Hirota’s bilinear equation has a linear combination of quadratic function and exponential function, we have eight classes of lump–kink solutions and four classes of lump–soliton solutions. Two illustrative examples of these mixed solutions are studied and their dynamics are analyzed.

Investigation of lump, soliton, periodic, kink, and rogue waves to the time-fractional phi-four and (2+1) dimensional CBS equations in mathematical physics

In this article, we investigate the lump, soliton, periodic, kink, and rogue waves to the time-fractional phi-four and (2+1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equations. The (G′/G,1/G)-expansion technique is applied to obtain the solutions of the phi-four and Calogero-Bogoyavlanskil schilf (CBS) equations with the contribution of conformable derivative. The stated method declines a huge volume of calculation. The (G′/G,1/G)-expansion technique is also known as the two-variable method. Consequently, lump, mixed lump, soliton, bell-shaped soliton, periodic, kink, periodic-kink, singular kink, rogue wave solutions are displayed in hyperbolic and trigonometric function solutions. Varieties of other traveling wave solutions are also formulated with the mentioned method. Moreover, for the physical explanation of the obtained solutions, the three-dimensional (3D) surface plots, contour plots combined with 3D surface plots, and two-dimensional (2D) plots are presented using the computer software MATLAB. Additionally, it is possible to deduce that the obtained solutions and their physical properties will aid in the understanding of water wave circulation in nonlinear dynamics.

Exact and explicit traveling wave solution to the time-fractional phi-four and (2+1) dimensional CBS equations using the modified extended tanh-function method in mathematical physics

This current study’s primary aim is to discover new and exact traveling wave solutions to the time-fractional phi-four equation and the (2+1) dimensional Calogero–Bogoyavlanskil schilf (CBS) equation in the perspective of nonlinear traveling wave phenomena. The modified extended tanh-function method is imposed on the phi-four and the (2+1) dimensional CBS equations in this case. Consequently, lump, mixed lump, lump-periodic, lump-periodic-kink, kink, singular kink, kink soliton, periodic, and singular solutions are exhibited in trigonometric, hyperbolic, and rational function solutions. To enucleate, the underlying traveling​ structures, achieved solutions are established by making their dynamic comportment of the exact solutions presented in three-dimensional (3D), contour, and two-dimensional (2D) chart with computational software MATLAB. In terms of conformable derivative, fractional traveling wave transformation, and the applied procedure, all the exact solutions obtained are considered to be novel. To comprehend the physical processes, we have portrayed the figures of the estimated solutions.

Nonautonomous lump solutions for a variable–coefficient Kadomtsev–Petviashvili equation

Based on the Hirota bilinear method, this paper investigates the nonautonomous lump solutions and mixed lump–stripe soliton solutions of a variable–coefficient Kadomtsev–Petviashvili equation. The displacements, trajectories and velocities of the nonautonomous lump wave are investigated by taking the variable–coefficient as the simplest power function, exponential function and sine function, respectively. In addition, the mixed lump–stripe soliton solutions are found to have periodic fission and fusion phenomena.

Abundant lump-type solutions for the extended (3+1)-dimensional Jimbo–Miwa equation

In this work, we research the mixed lump–kink solutions and their dynamic properties of the extended (3+1)-dimensional Jimbo–Miwa equation. Mixed lump–kink solutions are triggered by the interaction between lump soliton and exponential function, the interaction between lump soliton and hyperbolic function, and the interaction between lump soliton and periodic wave. Their dynamics properties are shown by means of many graphics and corresponding density plots.

Investigation of solitons and mixed lump wave solutions with [formula omitted]-dimensional potential-YTSF equation

The article investigates the exact and mixed lump wave solitons to the (3+1)-dimensional potential YTSF equation, which is an extension of the Bogoyavlenskii-Schif equation. Different types of interaction solutions in terms of a new merge of positive quadratic functions, trigonometric functions and hyperbolic functions are obtained, which are investigated using the extended three soliton test approach. The dynamical behavior of solutions has also depicted in different 3D representations. The representations show that the soliton solutions are obtained in the form of train and cusp. Some other solutions like lamb wave with high peak, and lump solutions with kink background are also observed in these representations.

Mixed lump and soliton solutions for a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation

Under investigation is a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation which can be used to describe nonlinear wave propagation in dissipative media. Via the bilinear transformation method, the mixed lump and soliton solutions are obtained for the equation. The asymptotic behavior of the mixed solutions are analyzed. Furthermore, the fusion and fission behaviors of the lump and soliton are observed for the first time. The lump and soliton can merge into a whole soliton over time, or, on the contrary, the soliton may differentiate into a lump and a new soliton. During the processes, the amplitude of the lump will greatly vary, while the amplitude of the soliton will change slightly.

Integrability aspects and localized wave solutions for a new $$\mathbf (4+1) $$-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper, we introduce a new integrable nonlinear evolution equation in $$4+1$$ dimensions, which is an extension of Boiti–Leon–Manna–Pempinelli equation. We prove that this new equation has the Painleve property. By using the Bell polynomial approach, we obtain the bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws. Furthermore, several types of new exact solutions are also constructed based on the Hirota bilinear method, including the N-soliton solutions, periodic soliton solutions and mixed lump–kink wave solutions. The dynamics and interactions of localized wave solutions are illustrated by some graphs.

Non-traveling lump solutions and mixed lump–kink solutions to (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation

Through Hirota bilinear form and symbolic computation with Maple, we investigate some non-traveling lump and mixed lump–kink solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Caudrey–Doddy–Gibbon–Kotera–Sawada equation by an extended method. Firstly, the non-traveling lump solutions are directly obtained by taking the function [Formula: see text] as a quadratic function. Secondly, we can get the interaction solutions for a lump solution and one kink solution by taking [Formula: see text] as a combination of quadratic function and exponential function. Finally, the interaction solutions between a lump solution and a pair of kinks solution can be derived by taking [Formula: see text] as a combination of quadratic function and hyperbolic cosine function. The dynamic phenomena of the above three types of exact solutions are demonstrated by some figures.

Mixed lump–soliton solutions to the two-dimensional Toda lattice equation via symbolic computation

Based on symbolic computation, we derive explicit rational and rational–exponential solutions to two-dimensional Toda lattice equation via the direct bilinear method. For two types of solutions, the auxiliary functions are taken as the ansatzs with the quadratic function and the combination of quadratic and exponential functions, respectively. It is shown that the purely rational solution depicts a lump with the locality in all directions in the space. The mixed rational–exponential solutions are classified to two kinds of interaction solution: (i) The first one describes the interaction of one lump with one line soliton, in which fission and fusion phenomena occurs between both local waves. (ii) The second one corresponds to the interaction of one lump with two resonant line solitons, which indicates a rogue wave arising from the resonant soliton pair.

Mixed lump–stripe soliton solutions to a dimensionally reduced generalized Jimbo–Miwa equation

With the help of the direct bilinear method, explicit rational and rational–exponential solutions to a dimensionally reduced generalized Jimbo–Miwa equation are derived by assuming the auxiliary function as the quadratic function and the combinations of the quadratic function and the exponential one. It is shown that three kinds of exact solutions describe the lump, the interaction consisting of one lump and one stripe soliton, and the rogue wave arising from the stripe soliton pair respectively. Local characteristics and interaction properties for these mixed nonlinear waves are analyzed in detail.

Mixed lump–soliton solutions of the (3+1)-dimensional soliton equation

Many mixed lump–soliton solutions for the (3+1)-dimensional integrable equation are obtained by using the Hirota bilinear method. These mixed lump–soliton solutions with arbitrary constants are given out analytically and graphically.

Fission and fusion interaction phenomena of mixed lump kink solutions for a generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this paper, a generalized (3[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili (gBKP) equation is investigated by using the Hirota’s bilinear method. With the aid of symbolic computation, some new lump, mixed lump kink and periodic lump solutions are derived. Based on the derived solutions, some novel interaction phenomena like the fission and fusion interactions between one lump soliton and one kink soliton, the fission and fusion interactions between one lump soliton and a pair of kink solitons and the interactions between two periodic lump solitons are discussed graphically. Results might be helpful for understanding the propagation of the shallow water wave.

Construction of lump soliton and mixed lump stripe solutions of (3+1)-dimensional soliton equation

Abstract In this letter, we apply two different ansatzs for constructing the lump soliton and mixed lump strip solutions of (3+1)-dimensional soliton equation, which is associating with the Hirota bilinear form. These lump soliton solutions rationally localized in all directions in the space. The solutions of interactions between a lump and a stripe are shown by graphic illustration of some special solutions which would give us a better understanding on the evolution of solutions of waves.

Mixed lump–kink solutions to the KP equation

By using the Hirota bilinear form of the KP equation, twelve classes of lump–kink solutions are presented under the help of symbolic computations with Maple. Analyticity is naturally achieved by taking special choices of the involved parameters to guarantee a positive constant term. A key step in generating lump–kink solutions is to combine quadratic functions and the exponential function in the second-order logarithmic derivative transformation.