Binary Bell Polynomials Research Articles

Soliton dynamics in (2+1) dimensional Heisenberg spin chain with Dzyaloshinskii–Moriya interaction in nanowire systems

In this study, we delve into the theoretical framework of the (2+1)-dimensional Kadomtsev–Petviashvili equation coupled with Dzyaloshinskii–Moriya (DM) interaction, elucidated from the Landau-Lifshitz equation, a prominent model in the realm of ferromagnetic dynamics. Employing the binary Bell polynomial technique along with the Hirota bilinear form, we construct one- and two-soliton solutions for the considered physical system. Through the manipulation of parameters present in these solutions, we explore the various nonlinear dynamical phenomena characterizing ferromagnetic nano-wires with DM interaction. Our investigations elucidate that solitons can be effectively manipulated in the nanowire system through judicious selection of system parameters. Moreover, our findings illustrate that soliton profiles exhibit compression, amplification, shifting, and changes in orientation during evolution. These insights hold significant implications for experimentalists aiming to analyze the propagation of solitons in ferromagnetic nanowire systems.

Lax integrability and nonlinear dispersive wave phenomenon for the (3 + 1) dimensional Kudryashov–Sinelshchikov equation

A (3 + 1) dimensional Kudryashov–Sinelshchikov equation is investigated in this paper, which describes bubbles in the liquid fluctuations. By virtue of the binary Bell polynomials, the bilinear representation, bilinear Bäcklund transformation with associated Lax pair are obtained, respectively. Moreover, utilizing Hirota’s bilinear representation, four new lump solutions are constructed and the interaction phenomenon between lump and periodic solution is thoroughly examined. The work also illustrates the intriguing dynamical behavior with the aid of Maple software, which plots the three-dimensional surface, two-dimensional density, and contour profiles of the solutions constructed in this work in various planes.

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials

Abstract In this article, the (2+1)-dimensional KdV equation by Hirota’s bilinear scheme is studied. Besides, the binary bell polynomials and then the bilinear form is created. In addition, an interaction lump with k k -soliton solutions of the addressed system with known coefficients is presented. With the assistance of the stated methodology, a cloaked form of an analytical solution is discovered in expressions of lump-soliton rational functions with a few lovable parameters. Solutions to this study’s problems are identified specifically as belonging to the lump-one, two, three, and four soliton solutions. By defining the specific advantages of the epitomized parameters by the depiction of figures and by interpreting the physical occurrences are established acceptable soliton arrangements and dealt with the physical importance of the obtained arrangements. Finally, under certain conditions, the physical behavior of solutions is analyzed by using the mentioned method. Moreover, the graphs with high resolutions including three-dimensional plots, density plots, and two-dimensional plots to determine a deep understanding of plotted solutions that will arise in the applied mathematics and nonlinear physics are employed.

Multi-solitons and integrability for a (2+1)-dimensional variable coefficients Date–Jimbo–Kashiwara–Miwa equation

Under investigation in this paper is a (2+1)-dimensional generalized variable coefficients Date–Jimbo–Kashiwara–Miwa equation, which describes the nonlinear dispersive wave in inhomogeneous media. Via the generalized Laurent series truncated at the constant-level term, an auto-Bäcklund transformation is derived. Bilinear forms, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota method.

Newly exploring the Lax pair, bilinear form, bilinear Bäcklund transformation through binary Bell polynomials, and analytical solutions for the (2 + 1)-dimensional generalized Hirota–Satsuma–Ito equation

The (2+1)-dimensional generalized Hirota–Satsuma–Ito equation describing the numerous wave dynamics in shallow waters is investigated in this study. The integrable characteristics of the aforesaid equation, such as a bilinear Bäcklund transformation and Lax pair, are revealed using the Bell polynomials method. First, using this technique, with the aid of Hirota operators, the bilinear form is constructed for the considered equation. In addition, the bilinear Bäcklund transformation and the Lax pair of the aforesaid equation are derived successfully using the bilinear form. Moreover, the bilinear form is also used to construct analytical solutions utilizing the three-wave approach with a test function. While using this method, numerous analytical solutions are derived, which are not presented in the literature. A three-dimensional graph has been plotted for each of the obtained results by giving the appropriate values of the free parameters. These plots reveal a wide variety of wave behavior, such as kink-soliton, periodic wave, anti-kink soliton, and complex periodic wave solutions.

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.

Lax integrability and soliton solutions of the (2 + 1)- dimensional Kadomtsev– Petviashvili– Sawada–Kotera– Ramani equation

In this paper, a new (2 + 1)-dimensional nonlinear evolution equation is investigated. This equation is called the Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation, which can be seen as the two-dimensional extension of the Korteweg–de Vries–Sawada–Kotera–Ramani equation. By means of Hirota’s bilinear operator and the binary Bell polynomials, the bilinear form and the bilinear Bäcklund transformation are obtained. Furthermore, by application of the Hopf-Cole transformation, the Lax pair is also derived. By introducing the new potential function, infinitely many conservation laws are constructed. Therefore, the Lax integrability of the equation is revealed for the first time. Finally, as the analytical solutions, the N-soliton solutions are presented.

Bilinear form and soliton solutions for a higher order wave equation

Under investigation in this paper is a higher order wave equation of the Korteweg–de Vries type, which describes the propagation of more complex wave in the shallow water. Bilinear form is derived based on the binary Bell polynomials. One-soliton solutions are constructed via the Hirota method. Two-soliton solutions are tried to be constructed, but it degenerates to the one-soliton solutions. Influences of the coefficients α and β on the soliton velocity and amplitude are analyzed through the characteristic line.

Bifurcation, bilinear forms, conservation laws and soliton solutions of the temporal-second-order KdV equation

The temporal-second-order KdV equation, which describes the propagation of two wave modes with different phase velocities and same dispersion relation, nonlinearity and dispersion parameters are investigated. The similarity reductions and new exact solutions are obtained via the Kudryashov method and a new version of Kudryashov method. Furthermore, the conservation laws are derived using the new conservation theorem. The bilinear forms and bilinear Bäcklund transformation of the temporal-second-order KdV equation are derived through the binary Bell polynomial. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota method. The characteristics and interaction of the solitons are discussed graphically. We discuss the effect of the phase velocities [Formula: see text] and [Formula: see text] and the parameters of nonlinearity [Formula: see text] and [Formula: see text] on the soliton amplitudes and velocities. Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the temporal-second-order KdV equation.

Integrability and high-order localized waves of the (4 + 1)-dimensional nonlinear evolution equation

Constructing the exact solutions of high-dimensional nonlinear evolution equations and exploring their dynamics have always been important and open problems in real-world applications. The celebrated Korteweg-de Vries equation [KdV] and Kadomtsev-Petviashvili [KP] equation are typical examples of one-dimensional and two-dimensional integrable equations respectively. A natural idea is to investigate the integrable analogues of these equations in higher dimensional space. In this paper, an integrable extension of the Kadomtsev-Petviashvili equation, the (4 + 1)-dimensional nonlinear evolution equation (NLEE) [Physical Review Letters 96, (2006) 190201], is investigated. By dimensionality reduction we obtain three new nonlinear equations, namely (3 + 1)-dimensional NLEE, (2 + 1)-dimensional NLEE and (1 + 1)-dimensional NLEE. Bäcklund transformations and multi-soliton solutions are important characteristics of integrable equations. Based on the binary Bell polynomials and Hirota bilinear method we derive bilinear Bäcklund transformations and N-soliton solutions of these new equations, which show that these equations are integrable. We also give the multiple rational solutions of these new equations. These new integrable NLEEs enrich the models of integrable systems and help understand the new characteristics of nonlinear dynamics in real-world applications.

Integrability, conservation laws and exact solutions for a model equation under non-canonical perturbation expansions

In this paper, the non-linear for the small long amplitude waves in two dimensional (2D) shallow water waves propagation with free surface are considered. The shallow water wave problem leads to the non-linear Hamiltonian model equation. Based on the binary Bell-polynomials approach, the bilinear form, bilinear Bäcklund transformation and multiple wave solutions are obtained. The conservation laws are constructed using two different techniques, namely, the Ibragimov's theorem and the multiplier method. The Noether's approach was applied to the non-linear Hamiltonian model equation to obtain the conservation laws. Also, we show that the non-linear Hamiltonian model equation is nonlinearly self-adjoint. Conserved quantities of Hamiltonian model equation are illustrated. Finally, with the help of the extended homogeneous balance method, and an exponential method, a set of new exact solutions for the non-linear Hamiltonian model equation are obtained.

Combined Damped Sinusoidal Oscillation Solutions to the (3 + 1)-D Variable-Coefficient Generalized NLW Equation in Liquid with Gas Bubbles

This paper investigates the combined damped sinusoidal oscillation solutions to the 3 + 1 -D variable-coefficient (VC) generalized nonlinear wave equation. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize a binary Bell polynomial transformation for reducing the Cole-Hopf algorithm to get the exact solutions of the VC generalized NLW equation. The damped sinusoidal oscillations for two cases of the nonlinear wave ordinary differential equation will be studied. Using suitable mathematical assumptions, the novel kinds of solitary, periodic, and singular soliton solutions are derived and established in view of the trigonometric and rational functions of the governing equation. To achieve this, the illustrative example of the VC generalized nonlinear wave equation is provided to demonstrate the feasibility and reliability of the procedure used in this study. The trajectory solutions of the traveling waves are shown explicitly and graphically. The effect of the free parameters on the behavior of acquired figures of a few obtained solutions for two nonlinear rational exact cases was also discussed. By comparing the proposed method with the other existing methods, the results show that the execution of this method is concise, simple, and straightforward.

Painlevé analysis, integrability property and multiwave interaction solutions for a new (4+1)-dimensional KdV–Calogero–Bogoyavlenkskii–Schiff equation

In this letter a novel (4+1)-dimensional KdV–Calogero–Bogoyavlenkskii–Schiff (KdV-CBS) equation is investigated. The Painlevé analysis shows that this higher dimensional nonlinear equation passes the integrability test. Its integrable properties, including bilinear Bäcklund transformation, Lax pair as well as N-soliton solution, are systematically derived using the binary Bell polynomial method. Furthermore, incorporating the inheritance solving technique into direct algebraic method, we construct the multiwave solutions of 1-lump and (N−2)-soliton (N>2). Multiple choices of parameters in the obtained solutions lead to rich interactions among lump waves, kink waves and breather waves. Several interesting interactions of the multiwave solutions are shown by graphs.

N‐Lump to the (2+1)‐Dimensional Variable‐Coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

In this research, the (2 + 1)‐dimensional (D) variable‐coefficient (VC) Caudrey–Dodd–Gibbon–Kotera–Sawada model used in soliton hypothesis and implemented by operating the Hirota bilinear scheme is studied. A few modern exact analytical outcomes containing interaction between a lump‐two kink soliton, interaction between two‐lump, the interaction between two‐lump soliton, lump‐periodic, and lump‐three kink outcomes for the (2 + 1)‐D VC Caudrey–Dodd–Gibbon–Kotera–Sawada equation by Maple Symbolic packages are obtained. By employing Hirota’s bilinear technique, the extended soliton solutions according to bilinear frame equation are received. For this model, the contemplated model can be got by multi‐D binary Bell polynomials (bBPs). In addition, the analytical analysis of the high‐order soliton outcomes to present the discipline of outcomes. The effect of the free parameters on the behavior of acquired figures of a few obtained solutions for the nonlinear rational exact cases was also discussed. The above technique could also be employed to get exact solutions for other nonlinear models in physics, applied mathematics, and engineering.

Lump and Interaction Solutions to the ( 3 + 1 )-Dimensional Variable-Coefficient Nonlinear Wave Equation with Multidimensional Binary Bell Polynomials

In this paper, we study the ( 3 + 1 )-dimensional variable-coefficient nonlinear wave equation which is taken in soliton theory and generated by utilizing the Hirota bilinear technique. We obtain some new exact analytical solutions, containing interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the generalized ( 3 + 1 )-dimensional nonlinear wave equation in liquid with gas bubbles by the Maple symbolic package. Making use of Hirota’s bilinear scheme, we obtain its general soliton solutions in terms of bilinear form equation to the considered model which can be obtained by multidimensional binary Bell polynomials. Furthermore, we analyze typical dynamics of the high-order soliton solutions to show the regularity of solutions and also illustrate their behavior graphically.

Bäcklund transformation and multi-soliton solutions for the discrete Korteweg–de Vries equation

In this paper, a discrete Korteweg–de Vries equation for an LC network consisting of the voltage-dependent capacitors and current-dependent inductors is investigated by symbolic computation. Binary Bell polynomials are applied to the discrete Korteweg–de Vries equation which is reduced to the corresponding bilinear form directly rather than transformed into its fourlinear one first. Simultaneously, the dependent variable transformation is acquired through the deriving procedures. Bäcklund transformation of the dKdV equation is derived with the introduced mixing variables instead of the exchange formulae. Solitonic propagation and interaction in the electrical circuit are illustrated and analyzed.

Bilinear Bäcklund transformation, N‐soliton, and infinite conservation laws for Lax–Kadomtsev–Petviashvili and generalized Korteweg–de Vries equations

In this paper, we obtain the bilinear form for the Lax–Kadomtsev–Petviashvili (Lax–KP) and the generalized (3 + 1)‐dimensional Korteweg–de Vries equations based on the binary Bell polynomials. Accordingly, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws will be constructed to Lax–KP and the generalized (2 + 1)‐dimensional Korteweg–de Vries equation . At the same time, we get another bilinear Bäcklund transformation. Finally, exact solutions were obtained by using the exchange formulas for Hirota's bilinear operators.

CHARACTERISTICS OF NEW TYPE ROGUE WAVES AND SOLITARY WAVES TO THE EXTENDED (3+1)-DIMENSIONAL JIMBO-MIWA EQUATION

<abstract> On the basis of the binary Bell polynomial scheme, the bilinear form of the extended (3+1)-dimensional Jimbo-Miwa (JM) equation, is constructed. Then, a class of new type rogue waves solutions to the extended (3+1)-dimensional JM equation, is found. It mainly includes the lump solutions, lumpoff solutions and instanton solutions. Their nonlinear evolutionary processes by 3D- and 2D-graphs, are shown. Finally, a direct method which is called the tanh-function method was used to get solitary waves of this considered model. These results can help us better understand interesting physical phenomena and mechanism. </abstract>

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation* *Project supported by the National Natural Scinece Foundation of China (Grant Nos. 11671219, 11871446, 12071304, and 12071451).

Within the (2 + 1)-dimensional Korteweg–de Vries equation framework, new bilinear Bäcklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function ϕ(y), a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method. By choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of the obtained solitons. Additionally, two-dimensional (2D) rogue waves (localized in both space and time) on the soliton plane are presented, we refer to them as deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane, and its evolution process is analyzed in detail. The deformed 2D rogue wave solutions are constructed successfully, which are closely related to the arbitrary function ϕ(y). This new idea is also applicable to other nonlinear systems.