Lump-kink Solutions Research Articles

Dynamics of lump solutions, lump-kink solutions and periodic lump solutions in a (3+1)-dimensional generalized Jimbo–Miwa equation

Under investigation in this paper is a (3+1)-dimensional generalized Jimbo–Miwa equation(gJM), which describes many interesting three dimensional nonlinear waves in physics. Based on the properties of the Bell's polynomial, a very classic method is presented to find the Hirota bilinear form of the equation. By using the resulting bilinear form, we systematically construct its lump solutions and lump-like solutions including lump-kink solutions and periodic lump solutions with their corresponding limitations. Moreover, we show that the conditions satisfying by lump solutions are considered to satisfy several important properties, including localization, positive and analyticity. The interaction solutions among a lump and a kink wave are also provided by considering a special function. The corresponding interaction phenomenon does not possess elasticity, but has a constantly removing process between a lump and a kink wave. Finally, periodic lump solutions are also derived by employing the three-wave method. It is hoped that our results can help rich the dynamic behaviors of the (3+1)-dimensional Jimbo–Miwa-like equations.

Lump, mixed lump-kink, breather and rogue waves for a B-type Kadomtsev-Petviashvili equation

ABSTRACT The B-type Kadomtsev-Petviashvili (BKP) equations can describe certain nonlinear phenomena in the fluids. In this paper, a BKP equation is investigated. Lump-wave, mixed lump-kink wave, breather-wave and rogue-wave solutions are constructed via the symbolic computation and Hirota method. Kink-shaped traveling wave solutions are derived via the polynomial expansion method. According to the mixed lump-kink wave solutions, we graphically analyze the interactions between the lump wave and kink wave and observe that (1) after the fission of the kink wave, the kink wave splits into one kink wave and one lump wave; (2) after the fusion between the lump and kink waves, the lump and kink merge together. Besides, we discuss the fission-fusion phenomenon between the lump wave and a pair of the kink waves and find that the higher kink wave splits into a kink wave and a lump wave, and then the lump wave and the lower kink wave merge together. Breather waves are displayed, based on which we construct the rogue waves with the periods of the breather waves becoming the infinity.

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.

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.

M-lump and interactive solutions to a (3 $${+}$$ + 1)-dimensional nonlinear system

This paper aims at computing M-lump solutions for the $$(3+1)$$ -dimensional nonlinear evolution equation. These solutions in all directions decline to an identical state obtained by employing the “long wave” limit with respect to the N-soliton solutions which are got by using the direct methods. Subsequently, we discuss the dynamic properties of the M-lump solutions which describe the multiple collisions of lumps. Based on the obtained lump solutions, the lump–kink solutions are also obtained. In addition, the periodic interactive solutions are given.

Abundant lump and lump–kink solutions for the new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

By utilizing the Hirota’s bilinear form and symbolic computation, abundant lump solutions and lump–kink solutions of the new (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation are derived in this work. Meanwhile, the interaction between lump solutions and the exponential function is also investigated. The dynamic properties of these obtained lump and interaction solutions are analyzed and described in figures by selecting appropriate parameters.

Mixed lump-kink and rogue wave-kink solutions for a (3 + 1) -dimensional B-type Kadomtsev-Petviashvili equation in fluid mechanics

Under investigation is a (3 + 1) -dimensional B-type Kadomtsev-Petviashvili equation, which describes the weakly dispersive waves in a fluid. Via the Hirota method and symbolic computation, we obtain the mixed lump-kink and mixed rogue wave-kink solutions. Through the mixed lump-kink solutions, we observe three different phenomena between a lump and one kink. For the fusion phenomenon, a lump and a kink are merged with the lump’s energy transferring into the kink gradually, until the lump merges into the kink completely. Fission phenomenon displays that a lump separates from a kink. The last phenomenon shows that a lump travels together with a kink with their amplitudes unchanged. In addition, we graphically study the interaction between a rogue wave and a pair of the kinks. It can be observed that the rogue wave arises from one kink and disappears into the other kink. At certain time, the amplitude of the rogue wave reaches the maximum.

Diversity of interaction solutions to the (2+1)-dimensional Ito equation

We aim to show the diversity of interaction solutions to the (2+1)-dimensional Ito equation, based on its Hirota bilinear form. The proof is given through Maple symbolic computations. An interesting characteristic in the resulting interaction solutions is the involvement of an arbitrary function. Special cases lead to lump solutions, lump-soliton solutions and lump-kink solutions. Two illustrative examples of the resulting solutions are displayed by three-dimensional plots and contour plots.

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.

Mixed lump-kink solutions to the BKP equation

By using the Hirota bilinear form of the (2+1)-dimensional BKP equation, ten classes of interaction solutions between lumps and kinks are constructed through Maple symbolic computations beginning with a linear combination ansatz. The resulting lump-kink solutions are reduced to lumps and kinks when the exponential function and the quadratic function disappears, respectively. Analyticity is naturally guaranteed for the presented lump-kink solution if the constant term is chosen to be positive.