Resonant Y-shape soliton, X-shape soliton, breather wave and abundant travelling wave solutions to the generalised (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

The (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation with weak nonlinearity, dispersion and perturbation can denote the development of the long waves and the surface waves in fluid dynamics. In this paper, the KP equation is illustrated with the symbolic computation. The mixed interaction solutions of local wave, solitary wave, breather wave, exploding wave and periodic wave for the equation are derived by the Hirota method. The effects of dispersion, nonlinearity and other parameters on the interactions are investigated. The solitary wave can be amplified via introducing the local wave. Adjusting the parameters can make the transmission of localized and breather wave more stable. Moreover, a new exploding and periodic wave is observed. It is useful for enriching the dynamic patterns of the wave solutions.

Integrability, breather, lump and quasi-periodic waves of non-autonomous Kadomtsev–Petviashvili equation based on Bell-polynomial approach

PurposeThe purpose of this paper is to study the breather waves, rogue waves and solitary waves of an extended (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation, which can be used to depict many nonlinear phenomena in fluid dynamics and plasma physics.Design/methodology/approachThe authors apply the Bell’s polynomial approach, the homoclinic test technique and Hirota’s bilinear method to find the breather waves, rogue waves and solitary waves of the extended (3 + 1)-dimensional KP equation.FindingsThe results imply that the extended (3 + 1)-dimensional KP equation has breather wave, rogue wave and solitary wave solutions. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior.Originality/valueThese results may help us to further study the local structure and the interaction of solutions in KP-type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of such equations.

Resonant line wave soliton solutions and interaction solutions for (2+1)-dimensional nonlinear wave equation

On breather waves, rogue waves and solitary waves to a generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation

Under investigation is a generalized (3 + 1)-dimensional variable- coefficient Kadomtsev– Petviashvili equation in fluid mechanics. Various exact analytical solutions are obtained by Hirota’s bilinear method, such as lump-type, breather wave and kink-solitary wave solutions. We discuss the interaction between lump wave and solitary waves, and the interaction between lump wave and periodic wave. The physical structure and propagation characteristics of obtained solutions are shown by some 3D graphics.

Based on the hybrid solutions to (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation, the motion trajectory of the solutions to KP equation is further studied. We obtain trajectory equation of a single lump before and after collision with line, lump, and breather waves by approximating solutions of KP equation along some parallel orbits at infinity. We derive the mathematical expression of the phase change before and after the collision of a lump wave. At the same time, we give some collision plots to reveal the obvious phase change. Our method proposed to find the trajectory equation of a lump wave can be applied to other (2+1)-dimensional integrable equations. The results expand the understanding of lump, breather, and hybrid solutions in soliton theory.

Dynamics of the breather waves, rogue waves and solitary waves in an extend Kadomtsev–Petviashvili equation

On the solitary waves, breather waves and rogue waves to a generalized ([formula omitted])-dimensional Kadomtsev–Petviashvili equation

The ([Formula: see text])-dimensional B-type Kadomtsev–Petviashvili (BKP) equation is utilized to depict weakly dispersive waves propagating in the fluid mechanics. According to [Formula: see text]-soliton solutions, resonance Y-shaped soliton and its interaction with other local wave solutions of the ([Formula: see text])-dimensional BKP equation are derived by introducing the constraint conditions. These types of hybrid soliton solutions exhibit the complex interaction phenomenon among resonance Y-shaped solitons, breather waves, line solitary waves and high-order lump waves. The dynamic behaviors of such interaction solutions are analyzed and illustrated.

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.

In this paper, we incorporate new constrained conditions into N-soliton solutions for a (2+1)-dimensional fourth-order nonlinear equation recently developed by Ma, resulting in the derivation of resonant Y-type solitons, lump waves, soliton lines and breather waves. We utilize the velocity-module resonance method to mix resonant waves with line waves and breather solutions. To investigate the interaction between higher-order lumps and resonant waves, soliton lines, and breather waves, we use the long wave limit method. We analyze the motion trajectory equations before and after the collision of lumps and other waves. To illustrate the physical behavior of these solutions, several figures are included. We also analyze the Painlevé integrability and explore the existence of multi-soliton solutions for the Ma equation in general. We demonstrate that our specific Ma-type equation is not Painlevé integrable; however, it does exhibit multi-soliton solutions.

Under investigation in this paper is a (3[Formula: see text]+[Formula: see text]1)-dimensional generalized Kadomtsev–Petviashvili equation, which describes the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid. Via the Hirota method and symbolic computation, the lump wave, breather wave and rogue wave solutions are obtained. We graphically present the lump waves under the influence of the dispersion effect, nonlinearity effect, disturbed wave velocity effects and perturbed effects: Decreasing value of the dispersion effect can lead to the range of the lump wave decreases, but has no effect on the amplitude. When the value of the nonlinearity effect or disturbed wave velocity effects increases respectively, lump wave’s amplitude decreases but lump wave’s location keeps unchanged. Amplitudes of the lump waves are independent of the perturbed effects. Breather waves and rogue waves are displayed: Rogue waves emerge when the periods of the breather waves go to the infinity. When the value of the dispersion effect decreases, range of the rogue wave increases. When the value of the nonlinearity effect or disturbed wave velocity effects decreases respectively, rogue wave’s amplitude decreases. Value changes of the perturbed effects cannot influence the rogue wave.

Under investigation in this paper is a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics and plasma physics. With the help of symbolic computation, we obtain and discuss the influence of the perturbed effect and disturbed wave velocity along the transverse spatial coordinate on the lump, lumpoff, rogue wave, breather wave and periodic lump solutions: When the value of decreases to −1, the amplitude of the lump wave becomes smaller; When the value of increases to 5, the location of the lump wave moves along the positive direction of the y (a transverse spatial coordinate) axis; When the value of decreases to 0.5, the location of the stripe soliton moves along the negative direction of the y axis and the amplitude of the lump wave becomes smaller; When the value of decreases to , the amplitude of the rogue wave becomes smaller; When the value of increases to 5, breather waves propagate along the positive t (the temporal coordinate) direction and distance between the adjacent crests becomes shorter; When the value of decreases to −1, breather waves propagate along the negative t direction and distance between the adjacent crests becomes shorter; When the value of decreases to 0.5, periodic lump waves move along the positive direction of the y axis. Lump solutions have more parameters than those in the existing literature. Lumpoff wave is generated from the process of the interaction between the lump wave and one stripe soliton. Moving path of the lumpoff wave is investigated via the moving path of the lump wave. Besides, we derive the rogue wave, breather wave and periodic lump solutions.

This paper presents a comprehensive investigation of an integrable (3+1)-dimensional pKP-BKP equation, a novel model that synthesizes the potential Kadomtsev-Petviashvili (pKP) and B-type Kadomtsev-Petviashvili (BKP) equations. The equation is highly applicable for modeling nonlinear wave phenomena, including oceanic surface waves and atmospheric disturbances, as it effectively captures the intricate interplay between dispersion and nonlinearity in multi-dimensional wave propagation. Using Hirota’s bilinear method, we first derive the three-soliton solution and analyze its nonlinear superposition mechanism and propagation dynamics, thereby identifying distinct interaction patterns. By strategically annihilating specific dispersion coefficients in the phase shifts, we construct resonant Y -type and X -type solitons. Through a systematic application of the long-wave limit method, we derive lump waves and characterize their dynamics, which are visualized using three-dimensional plots, time-evolving density maps, and trajectory projections. Breather wave solutions are systematically derived by introducing complex conjugate parameters, which reveals explicit amplitude-phase modulation relationships. Furthermore, we analyze the dynamics of localized interaction solutions among these waves, illustrated through dynamical evolution diagrams. This study enhances the understanding of complex nonlinear wave phenomena described by the pKP-BKP equation.