Pfaffian Solutions Research Articles

Superposition of soliton, breather and lump waves in a non-painlevé integrabale extension of the Boiti-Leon-Manna-Pempinelli equation

Abstract In this paper, we derive general $N$th-order Pfaffian solutions for a $(3+1)$-dimensional non-Painlev{'e} integrable extension of the Boiti-Leon-Manna-Pempinelli (BLMP) equation. Specifcally, we obtain $N$-soliton, higher-order breather, higher-order lump and hybrid solutions, and explore the superpositions of Y-shaped and X-shaped soliton-breather waves.
Moreover, we construct bilinear B"{a}cklund transformations, Lax pairs, and conservation laws using Bell polynomials.
Finally, we identify a similar equation in the literature and demonstrate that it represents another non-Painlevé integrable extension of the BLMP equation.

Pfaffian solutions and nonlinear dynamics of surface waves in two horizontal and one vertical directions with dispersion, dissipation and nonlinearity effects

This study delves into the exploration of the (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt (GFKDKK) system, a crucial nonlinear evolution equation governing wave motion across various physical domains. The core aim is to devise innovative analytical solutions for this system employing the Khater III technique and an enhanced Kudryashov methodology integrated with the conformable fractional derivative.The GFKDKK system holds pivotal significance in modeling and comprehending diverse scientific phenomena, ranging from fluid dynamics to plasma physics and nonlinear optics. This research endeavors to harness the potential of the Khater III technique and the updated Kudryashov approach to offer fresh analytical insights into nonlinear fractional differential equations, thus enriching the existing knowledge base in this domain.The attained solutions furnish invaluable insights into the dynamics of the system and its potential applications, thereby facilitating the development of effective numerical simulations and analytical frameworks. The study’s significance lies in its contribution to the ongoing exploration of nonlinear fractional differential equations and their practical implications across various industries.In conclusion, the research affirms the efficacy of the Khater III method and the enhanced Kudryashov technique, complemented by the conformable fractional derivative, in obtaining analytical solutions for the (3+1)-dimensional GFKDKK system. The novelty lies in the application of these methodologies to this specific system, resulting in the discovery of new analytical solutions that enhance our understanding of its behavior and characteristics.

Studies on an extended (3+1)-dimensional variable-coefficient shallow water wave equation: Bilinear form, Pfaffian solutions and Solitonic interactions

Studies on an extended (3+1)-dimensional variable-coefficient shallow water wave equation: Bilinear form, Pfaffian solutions and Solitonic interactions

Bilinear form, auto-Bäcklund transformations, Pfaffian, soliton, and breather solutions for a (3 + 1)-dimensional extended shallow water wave equation

Study of the water waves remains central to fluid physics, ocean dynamics, and engineering. In this paper, a (3 + 1)-dimensional extended shallow water wave equation is investigated via symbolic computation. Bilinear form and two kinds of the bilinear auto-Bäcklund transformations with some solutions are given via the Hirota method. The Nth-order Pfaffian solutions are worked out by means of the Pfaffian technique, where N is a positive integer. N-soliton solutions are exported through the Nth-order Pfaffian solutions. By virtue of the asymptotic analysis, elastic and inelastic interactions between the two solitons on some periodic backgrounds are discussed. Interaction among the three solitons is illustrated graphically. The higher-order breather solutions are investigated via the complex parameter relation. Elastic and inelastic interactions between the two breathers on the periodic backgrounds are depicted graphically. Hybrid solutions consisting of the solitons and breathers are obtained. Interaction between the one soliton and one breather on a periodic background is presented.

Wronskian solutions and Pfaffianization for a (3 + 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma

In this paper, we investigate a (3 + 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili (GVCKP) equation in a fluid or plasma. The Nth-order Wronskian solutions for that equation are derived and proved under certain variable-coefficient constraints, where N is a positive integer. One-, two-, and three-soliton solutions in the Wronskian for that equation are given. By means of the Pfaffianization procedure, a coupled (3 + 1)-dimensional GVCKP system is constructed from that equation. Bilinear form for that coupled system is exported. Under certain variable-coefficient constraints, those Wronski-type and Gramm-type Pfaffian solutions for that coupled system are obtained and proved with the help of the Pfaffian identities.

Pfaffian solutions and nonlinear waves of a (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt system in fluid mechanics

Fluid mechanics is concerned with the behavior of liquids and gases at rest or in motion, where the nonlinear waves and their interactions are important. Hereby, we study a (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt system in fluid mechanics. We determine a bilinear form of that system via the Hirota method. Nth-order Pfaffian solutions are obtained via the Pfaffian technique and our bilinear form, where N is a positive integer. Based on the Nth-order Pfaffian solutions, we derive the N-soliton, higher-order breather, and hybrid solutions. Using those solutions, we present the (1) elastic interaction between the two solitary waves with a short stem, (2) elastic interaction between the two solitary waves with a long stem, (3) fission between the two solitary waves, (4) fusion between the two solitary waves, (5) one breather wave, (6) elastic interaction between the two breather waves, (7) fission between the two breather waves, (8) fusion among the one breather wave and two solitary waves, and (9) elastic interaction between the one breather wave and one solitary wave.

Bilinear form and Pfaffian solutions for a (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt system in fluid mechanics and plasma physics

Bilinear form and Pfaffian solutions for a (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt system in fluid mechanics and plasma physics

Pfaffian, soliton, breather and hybrid solutions for a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili equation in fluid mechanics

Pfaffian, soliton, breather and hybrid solutions for a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili equation in fluid mechanics

Pfaffian, soliton, hybrid and periodic-wave solutions for a ($$3+1$$)-dimensional B-type Kadomtsev–Petviashvili equation in fluid mechanics

Pfaffian, soliton, hybrid and periodic-wave solutions for a ($$3+1$$)-dimensional B-type Kadomtsev–Petviashvili equation in fluid mechanics

On the Equations of Poizat and Liénard

AbstractWe study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non-strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations.

Pfaffian, breather, and hybrid solutions for a (2 + 1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics

Fluid mechanics is seen as the study on the underlying mechanisms of liquids, gases and plasmas, and the forces on them. In this paper, we investigate a (2 + 1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics. By virtue of the Pfaffian technique, the Nth-order Pfaffian solutions are derived and proved, where N is a positive integer. Based on the Nth-order Pfaffian solutions, the first- and second-order breather solutions are obtained. In addition, Y-type and X-type breather solutions are constructed. Furthermore, we investigate the influence of the coefficients in the system on those breathers as follows: The locations and periods of those breathers are related to δ1, δ2, δ3, δ4, and δ5, where δc's (c=1,2,3,4,5) are the constant coefficients in the system. Moreover, hybrid solutions composed of the breathers and solitons are derived. Interactions between the Y/X-type breather and Y-type soliton are illustrated graphically, respectively. Then, we show the influence of the coefficients in the system on the interactions between the Y/X-type breather and Y-type soliton.

Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

A universal property in terms of Hirota differential operators is presented, with an aim to construct Hirota bilinear equations possessing Pfaffian formulations in (2+1)‐dimensions. It is then shown that there exist linear combination solutions of exponential waves to various extended Hirota bilinear equations. Applications are made for the (2+1)‐dimensional generalized Hirota–Satsuma–Ito (HSI) equation, the special fourth‐order (2+1)‐dimensional nonlinear equation, and the fifth‐order Korteweg–de Vries (KdV) equation, thereby presenting their particular Pfaffian and multiple wave solutions.

Hybrid solutions for the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics

In fluid mechanics, the higher-dimensional and higher-order equations are constructed to describe the propagations of nonlinear waves. In this paper, we investigate the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. The Nth-order Pfaffian solutions are constructed and proved via the modified Pfaffian technique. The higher-order soliton, first- and second-order breather solutions are constructed based on the Nth-order Pfaffian solutions. We graphically demonstrate that the amplitudes and velocities of the solitons are affected by some variable coefficients. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breathers, lumps and solitons included in the hybrid solutions possess the same properties as those of the breather and lump solutions.

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.

Solitons and breather waves for the generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics, ocean dynamics and plasma physics

Abstract Under investigation in this paper is the (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system, which can be used to describe certain situations in fluid mechanics, ocean dynamics and plasma physics. The Nth-order Pfaffian and Wronskian solutions are derived via the Pfaffian and Wronskian techniques, respectively, where N is a positive integer. Asymptotic analysis implies that the interaction between the two solitons is elastic with certain conditions. Furthermore, we obtain the breather waves according to the extended homoclinic test technique. Propagation of the breather waves indicates that the breather waves can evolve periodically along a straight line with a certain angle with the x and y axes, and their wave lengthes, amplitudes and velocities remain unchanged during the propagation.

Solitons and periodic waves for the (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

Fluid mechanics has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering. Under investigation in this paper is the ($$2+1$$)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. Via the Pfaffian technique and certain constraint on the real constant $$\alpha $$, the Nth-order Pfaffian solutions are derived. One- and two-soliton solutions are obtained via the Nth-order Pfaffian solutions. Based on the Hirota–Riemann method, one- and two-periodic wave solutions are constructed. With the help of the analytic and graphic analysis, we notice that: (1) of the one soliton, amplitude is irrelevant to $$\gamma $$, a real constant coefficient in the equation, velocity along the x direction is independent of $$\gamma $$, while velocity along the y direction is proportional to $$\gamma $$; (2) one soliton keeps its amplitude and velocity invariant during the propagation and total amplitude of the two solitons in the interaction region is lower than that of any soliton; (3) one-periodic wave can be viewed as a superposition of the overlapping solitary waves, placed one period apart; (4) periodic behaviors for the two-periodic wave exist along the x and y directions, respectively; (5) under certain limiting conditions, one-periodic wave solutions approach to the one-soliton solutions and two-periodic wave solutions approach to the two-soliton solutions.

Solitons for the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid via the Pfaffian technique

Fluids are common in nature, the study of which helps the design of the related industries. Under investigation in this letter is the (2[Formula: see text]+[Formula: see text]1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Pfaffian solutions have been obtained based on the Pfaffian technique with the assistance of the real auxiliary function [Formula: see text]. N-soliton solutions with [Formula: see text] are constructed, where y is the scaled space coordinate and [Formula: see text] is the steady stream function in the irrotational incompressible flow. Background shapes of the solutions are affected by [Formula: see text], but the structures of the solutions are affected by the derivative of the log terms in the solutions. Neighborhoods at the origins of the solitons are different in consequence of the different values of the real auxiliary parameter a. One- and two-soliton solutions are illustrated, which are the superpositions of the two kink solitons and different forms of [Formula: see text]. Interactions of the two solitons are presented, from which we see that the velocities, amplitudes and shapes of the two solitons remain unchanged before and after each interaction.

Pfaffian and extended Pfaffian solutions for a (3+1)-dimensional generalized wave equation

Under investigation in this paper is a (3+1)-dimensional generalized wave equation, which is the second equation in the Kadomtsev–Petviashvili hierarchy. Linear partial differential conditions presented in this paper are different from the previous study. Pfaffian and extended Pfaffian solutions with a real function φ(y) have been obtained based on these linear partial differential conditions, where y is the scaled space coordinate, and compared with the solutions obtained in the previous articles, the advantage of our ones is that φ(y) could represent an inhomogeneous medium to influence or control the shape of waves. Specific solutions, namely the first- and second-order exact solutions from the Pfaffian and extended Pfaffian solutions are discussed. Discussions indicate that the first- and second-order exact solutions are of the same shape as the corresponding one- and two-soliton solutions on the x−t or z−t plane but of the different shape on the y−t plane.

Pfaffians of B-type Kadomtsev–Petviashvili equation and complexitons to a class of nonlinear partial differential equations in (3 $$+$$ + 1) dimensions

The aim of this paper is to investigate a class of generalised Kadomtsev–Petviashvili (KP) and B-type Kadomtsev–Petviashvili (BKP) equations, which include many important nonlinear evolution equations as its special cases. By applying the fundamental Pfaffian identity, a general Pfaffian formulation is established and all the involved generating functions for Pfaffian entries need to satisfy a system of combined linear partial differential equations. The illustrative examples of the presented Pfaffian solutions are given for the (3 $$+$$ 1)-dimensional generalised KP, Jimbo–Miwa and BKP equations. Moreover, we use the linear superposition principle to generate exponential travelling wave solutions and mixed resonant solutions of the considered equations.