Dimensional Generalized Kadomtsev Petviashvili Equation Research Articles

Dynamic analysis and multiple solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid

Dynamic analysis and multiple solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid

New exact solutions and related dynamic behaviors of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

New exact solutions and related dynamic behaviors of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

Wronskian solutions, bilinear Bäcklund transformation, quasi-periodic waves and asymptotic behaviors for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The N-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

Nonautonomous lump-periodic and analytical solutions to the ($$\varvec{3+1}$$)-dimensional generalized Kadomtsev–Petviashvili equation

Nonautonomous lump-periodic and analytical solutions to the ($$\varvec{3+1}$$)-dimensional generalized Kadomtsev–Petviashvili equation

Anomalous scattering of dark lumps to the (2+1)-dimensional generalized Kadomtsev–Petviashvili equation

The anomalous interactions of dark lumps that are completely different from the general lump collisions of the (2+1)-dimensional generalized Kadomtsev–Petviashvili equation were constructed. Different from the normal interaction mechanisms, lumps with same velocity and depth were obtained by extending the expression of rational solution based on Hirota’s Bilinear method. Moreover, interactions between dark lumps and soliton were also investigated. In particular, rogue wave excited from the lumps and stripe soliton in the interaction duration were observed. The dynamic behavior of dark lumps was analyzed in detail, and the results obtained were illustrated graphically.

A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions

Searching for higher-dimensional integrable models is one of the most significant and challenging issues in nonlinear mathematical physics. This paper aims to extend the classic lower-dimensional integrable models to arbitrary spatial dimension. We investigate the celebrated Kadomtsev–Petviashvili (KP) equation and propose its (n+1)-dimensional integrable extension. Based on the singularity manifold analysis and binary Bell polynomial method, it is found that the (n+1)-dimensional generalized KP equation has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, Bäcklund transformation as well as infinite conservation laws, and thus the (n+1)-dimensional generalized KP equation is proven to be completely integrable. Moreover, various types of localized solutions can be constructed starting from the N-soliton solutions. The abundant interactions including overtaking solitons, head-on solitons, one-order lump, two-order lump, breather, breather-soliton mixed solutions are analyzed by some graphs.

Residual symmetries, consistent-Riccati-expansion integrability, and interaction solutions of a new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation

With the aid of the Painlevé analysis, we obtain residual symmetries for a new (3+1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the (3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.

Shape-changed propagations and interactions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluids

In this article, we consider the (3+1)-dimensional generalized Kadomtsev–Petviashvili (GKP) equation in fluids. We show that a variety of nonlinear localized waves can be produced by the breath wave of the GKP model, such as the (oscillating-) W- and M-shaped waves, rational W-shaped waves, multi-peak solitary waves, (quasi-) Bell-shaped and W-shaped waves and (quasi-) periodic waves. Based on the characteristic line analysis and nonlinear superposition principle, we give the transition conditions analytically. We find the interesting dynamic behavior of the converted nonlinear waves, which is known as the time-varying feature. We further offer explanations for such phenomenon. We then discuss the classification of the converted solutions. We finally investigate the interactions of the converted waves including the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. And the mechanisms of the collisions are analyzed in detail. The results could enrich the dynamic features of the high-dimensional nonlinear waves in fluids.

Bilinear Bäcklund transformation, soliton and breather solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics

A (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid dynam-ics and plasma physics is hereby investigated. Via the Hirota method, bilinear Bäcklund transformation are obtained, along with two types of the analytic solutions. Kink-shaped soliton solutions are derived via the Hirota method. Breather solutions are derived via the extended homoclinic test approach and lump solutions are obtained from the breather solutions under a limiting procedure. We find that the shape and amplitude of the one-kink soliton keep unchanged during the propagation and the velocity of the one-kink soliton depends on all the coefficients in the equation. We graphically demonstrate that the interaction between the two-kink solitons is elastic, and analyse the solitons with the influence of the coefficients. We observe that the amplitudes and shapes of the breather and lump remain unchanged during the propagation, and graphically present the breathers and lumps with the influence of the coefficients in the equation.

Painlevé analysis, Bäcklund transformations and traveling-wave solutions for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation in a fluid

Fluids are seen in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation in a fluid. On the basis of the Painlevé analysis, we find that the equation is Painlevé integrable under a certain constraint. Through the truncated Painlevé expansion, we give an auto-Bäcklund transformation. By virtue of the Hirota method, we derive a bilinear auto-Bäcklund transformation. Via the polynomial-expansion method, traveling-wave solutions are obtained. We observe that the amplitude of a traveling wave remains invariant during the propagation. We graphically demonstrate that the amplitude of the traveling-wave is affected by the coefficients corresponding to the dispersion and nonlinearity effects, while other coefficients have no influence on the traveling-wave amplitude, which represent the perturbed effects and disturbed wave velocity effects.

Multiwave interaction solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation

To construct a class of new multiwave interaction solutions for the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation, we calculate different types of interaction solutions among solitons, periodic waves and rational waves using the direct algebraic method together with the inheritance solving skill. Moreover, a new algorithm is proposed with the aid of the simplified Hirota method, the conjugated parameters assignment and long wave limit strategies, from which multiwave interaction solutions among solitons, breathers and lump waves are generated.

Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions to the $$(3+1)$$-dimensional generalized Kadomtsev–Petviashvili equation

With the Hirota bilinear method and symbolic computation, we investigate the $$(3+1)$$ -dimensional generalized Kadomtsev–Petviashvili equation. Based on its bilinear form, the bilinear Backlund transformation is constructed, which consists of four equations and five free parameters. The Pfaffian, Wronskian and Grammian form solutions are derived by using the properties of determinant. As an example, the one-, two- and three-soliton solutions are constructed in the context of the Pfaffian, Wronskian and Grammian forms. Moreover, the triangle function solutions are given based on the Pfaffian form solution. A few particular solutions are plotted by choosing the appropriate parameters.

Breather, lump, shock and travelling-wave solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics and plasma physics

Investigation is conducted on a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics and plasma physics. By virtue of the homoclinic-test, ansatz and polynomial-expansion methods, we construct the breather and lump solutions, shock wave solutions and travelling-wave solutions, respectively. We observe that the breather propagates steadily along a straight line with certain angles with the x, y and z axes. We observe that the amplitude and shape of the bright-dark lump wave keep unchanged during the propagation. We observe that the amplitude and the shape of the shock wave keep unchanged during the propagation. We graphically analyse the effects of the coefficients in the equation on the breather, lump and shock waves. We find that the lump is the most stable while the shock wave is the least stable under the same perturbation among the breather, lump and shock waves.

Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general since they contain elliptic functions. Finally, we derive the conserved quantities of this equation by employing two approaches—the general multiplier approach and Ibragimov’s theorem. The importance of conservation laws is explained in the introduction. It should be pointed out that the investigation of higher dimensional nonlinear partial differential equations is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.

New types of interaction solutions to the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation

In this paper, with the aid of symbolic computation system Maple, and based on the simplified Hirota method and ansatz technique, we discussed the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation with [Formula: see text] to obtain lump solutions, lump–kink solutions and three classes of interaction solutions. Comparing our new results with other researchers’ results shows that using this method gives the more opportunity to solve the nonlinear partial differential equations that appear in mathematics, physics, biological engineering and other fields. We also presented profiles of new lump solution, lump–kink solutions and interaction solutions as illustrative examples.

Lump, lumpoff, rogue wave, breather wave and periodic lump solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics and plasma physics

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.

INTERACTION SOLUTIONS AND ABUNDANT EXACT SOLUTIONS FOR THE NEW (3+1)-DIMENSIONAL GENERALIZED KADOMTSEV-PETVIASHVILI EQUATION IN FLUID MECHANICS

In this work, we present the interaction solutions and abundant exact solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation based on the Hirota's bilinear form and a direct function. The obtained interaction solutions contain the interaction between the rational function and the $\tanh$ function and the interaction between the rational function and the $\cos$ function. The dynamical properties of these resulting solutions are analyzed and shown in three-dimensional plots, corresponding contour graphs and plane figures.

Lump and lump strip solutions to the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation

Based on symbolic computation, the lump and lump strip solutions of the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation are presented using the generalized bilinear operator with the parameter p = 3. Those solutions are obtained from polynomial solutions, and are simply classified into some classes. Three illustrative examples of the resulting solutions are displayed. Finally, the dynamic properties of the obtained solutions are described.

Mixed lump-stripe, bright rogue wave-stripe, dark rogue wave-stripe and dark rogue wave solutions of a generalized Kadomtsev–Petviashvili equation in fluid mechanics

Under investigation in this paper is a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics. We derive the mixed lump-stripe waves, bright mixed rogue wave-stripe, dark mixed rogue wave-stripe and dark rogue waves solutions by virtue of the symbolic computation. We observe the fission and fusion phenomena between the lump and one-stripe wave through the mixed-stripe wave solutions. Then, we observe that the influence of l1, l2, l3, l4, l5, l6, l7 and l8 on the mixed lump-stripe waves, where l1 and l2 represent the dispersion and nonlinear effects, l3, l6, l7 and l8 are the perturbed effects, while l4 and l5 stand for the disturbed wave velocities along the transverse spatial coordinates y and z, respectively. We graphically present the interaction between a rogue wave and a pair of stripe waves through the mixed rogue wave-stripe solutions. We derive a dark mixed rogue wave-stripe when l1 < 0. We study the influence of l1, l2, l3, l4, l5, l6, l7 and l8 on the rogue wave and a pair of stripe waves. We present the dark rogue wave with certain parameters and observe that two stripe waves merge into one stripe wave.

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.