Bell Polynomial Approach Research Articles

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

This article describes the infinite conservation law, quasi-periodic wave, breather, lump, and characteristic of integrability via bilinear Bäcklund and lax pairs of the non-autonomous Kadomtsev–Petviashvili (NKP) equation (in the presence of an external force and damping). Bell-polynomial is deduced from Bäcklund transformation from which infinite conservation law for NKP equation is derived. Obtaining well-known quasi-periodic solutions is an important aspect of this investigation. A limiting procedure is used for analyzing the asymptotic behavior of multi-periodic waves. This method proves rigorously that periodic waves tend toward soliton solutions under a small amplitude limit, which results in a relationship between periodic waves and soliton solutions. For the first time, the effect of external force in a quasi-periodic wave is demonstrated explicitly from the numerical understanding. Further, some complicated solution structures of the NKP equation such as lump and breather-type solitons are explored from the bilinear form of the said equation with the appropriate choice of polynomial functions. Important characteristics of lump and breather waves in different forcing backgrounds are graphically described. Finally, the lump wave of the NKP equation is explored from the breather wave in the limiting stage. It is also confirmed from the analytical results of the relevant motions that the velocity, maximum altitude, and interacting natures of the wave quantities are all influenced by the damping and forcing terms.

Integrability, Lax pair and solitary interactions of the non-isospectral (2+1)-dimensional Gardner system on directional surface waves with large spreading angles

The non-isospectral generalization of the celebrated (2+1)-dimensional Gardner system describes various nonlinear phenomena and the characteristics of oceanic waves, physical plasmas, solid-state materials, etc. This paper utilizes an extended Bell polynomial approach to analyze a large class of non-isospectral integrable systems systematically. We exert a concrete investigation on the integrability of the non-isospectral and variable-coefficient (2+1)-dimensional Gardner equation, which describes the directional surface waves with large spreading angles. We obtain the bilinear formalism, bilinear Bäcklund transformations, Lax pair and kink-type solitary interactions of the corresponding system. Moreover, with a periodic alternation on the system's coefficients, we graphically analyze the propagation and interaction between its non-isospectral solitary wave trains.

Integrability, bilinearization, solitons and exact three wave solutions for a forced Korteweg-de Vries equation

The Korteweg-de Vries (KdV) equation with a forcing term is considered which demonstrates the mathematical modelling of the physics of a shallow layer of fluid due to external forcing. The characteristic of integrability is investigated via Painlevè analysis and an auto Bäcklund transformation is fabricated via the truncated Painlevè expansion. Using the concept of Bell polynomial approach the bilinear Bäcklund transformation, Lax pair and infinite sequence of conservation laws are constructed. The one-soliton, two-soliton and three-soliton solutions are obtained via simplified Hirota method. We demonstrate the propagation and interaction of soliton solutions graphically. Moreover, a new form of ansatz in the exact three wave method (ETW) along with symbolic computation is applied and some new periodic type of three wave solutions which includes periodic two solitary wave solution, doubly periodic solitary wave solution and breather type of two solitary wave solution are obtained for the forced KdV equation.

Dynamics of abundant solutions to the generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

The present paper is devoted to discussion of (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation from point of view of their multi-soliton solutions and localized solutions associated with multi-soliton solutions. Firstly, taking advantage of the Bell-polynomial approach, we construct the Hirota bilinear form of (1.1). Based on that, the multi-soliton solutions are also singled out. Subsequently, the (3+1)-dimensional gBKP equation is also found to allow fruitful localized solutions, including breather, lump, rogue wave, and hybrid solutions. These results obtained in this work adequately illustrate the effectiveness of the long wave limit method and complex conjugate technique, which are expected to be employed to obtain more abundant exact solutions.

Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws

The (2+1)-dimensional Kadomtsev-Petviashvili type equations describe the nonlinear phenomena and characteristics in oceanography, fluid dynamics and shallow water. In the literature, a novel (2+1)-dimensional nonlinear model is proposed, and the localized wave interaction solutions are studied including lump-kink and lump-soliton types. Hereby, it is of further value to investigate the integrability characteristics of this model. In this paper, we firstly conduct the Painlevé analysis and find it fails to pass the Painlevé test due to a non-vanishing compatibility condition at the highest resonance level. Then we derive the soliton solutions and give the formula of the N-soliton solution, which is proved by means of the Hirota condition. The criterion for the linear superposition principle is also given to generate the resonant solutions. Bäcklund transformation, Lax pair and infinitely many conservation laws are derived through the Hirota bilinear method and Bell polynomial approach. As a result, we have a more overall understanding of the integrability characteristics of this novel (2+1)-dimensional nonlinear model.

The integrability of the coupled Ramani equation with binary Bell polynomials

Binary Bell polynomial approach is applied to study the coupled Ramani equation, which is the generalization of the Ramani equation. Based on the concept of scale invariance, the coupled Ramani equation is written in terms of binary Bell polynomials of two dimensionless field variables, which leads to the bilinear coupled Ramani equation directly. As a consequence, the bilinear Bäcklund transformation, Lax pair and conservation laws are systematically constructed by virtue of binary Bell polynomials.

Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.

Bell Polynomial Approach for the Solutions of Fredholm Integro-Differential Equations with Variable Coefficients

In this article, we approximate the solution of high order linear Fredholm integro-differential equations with a variable coefficient under the initial-boundary conditions by Bell polynomials. Using collocation points and t... | Find, read and cite all the research you need on Tech Science Press

Integrability aspects and localized wave solutions for a new $$\mathbf (4+1) $$-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper, we introduce a new integrable nonlinear evolution equation in $$4+1$$ dimensions, which is an extension of Boiti–Leon–Manna–Pempinelli equation. We prove that this new equation has the Painleve property. By using the Bell polynomial approach, we obtain the bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws. Furthermore, several types of new exact solutions are also constructed based on the Hirota bilinear method, including the N-soliton solutions, periodic soliton solutions and mixed lump–kink wave solutions. The dynamics and interactions of localized wave solutions are illustrated by some graphs.

Characteristics of integrability, bidirectional solitons and localized solutions for a ( $$3+1$$ 3 + 1 )-dimensional generalized breaking soliton equation

The characteristics of integrability, bidirectional solitons and localized solutions are investigated for a ( $$3+1$$ )-dimensional breaking soliton (GBS) equation with general forms. Firstly, starting from the GBS equation, we perform the singularity manifold analysis and obtain a new integrable model in the sense of Painleve property. Secondly, taking advantage of the Bell polynomial approach, we construct the Backlund transformation, Lax pair and an infinite sequence of conservation laws. Subsequently, this new equation is also found to allow bidirectional soliton solutions, and the head-on and overtaking collisions between solitons are illustrated by some illustrative graphs. Finally, some localized excitations, such as lump solution, multi-dromions, periodic solitary waves solution, are obtained.

Integrability, bilinearization and analytic study of new form of [formula omitted]-dimensional B-type Kadomstev–Petviashvili (BKP)- Boussinesq equation

Nonlinear and dispersive media deals with propagation of waves which are characterized by several nonlinear partial differential equations. A new form of (3+1) - dimensional B-type Kadomstev–Petviashvili - Boussinesq equation being one of them, has been investigated via bringing light on singularities with help of analysis of Painlevé property and it turns out that the equation clears the Painlevé test which affirms its explicit integration. Truncated Painlevé expansion and Bell polynomial approach is applied to establish bilinear equation. Furthermore by using the novel test function, various exact solutions consisting numerous arbitrary constants are revealed in an orderly way. Graphical representation and distinct properties are discussed corresponding to the decorum of each acquired solution. Various patterns, including kink-like along with periodic exhibiting solitons, are explored.

Rogue waves, homoclinic breather waves and soliton waves for a (3 + 1)-dimensional non-integrable KdV-type equation

PurposeThe purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion.Design/methodology/approachThe authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional non-integrable KdV-type equation. The used approach formally derives the essential conditions for these solutions to exist.FindingsThe results show that the equation exists rogue waves, homoclinic breather waves and soliton waves. To better understand the dynamic behavior of these solutions, the authors analyze the propagation and interaction properties of the these solutions.Originality/valueThese results may help to investigate the local structure and the interaction of waves in KdV-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Homoclinic breather waves, rogue waves and solitary waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation

PurposeThe purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.Design/methodology/approachThe authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.FindingsThe results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.Originality/valueThese results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Bell-polynomial approach and Wronskian determinant solutions for three sets of differential–difference nonlinear evolution equations with symbolic computation

Under investigation in this paper are the Belov–Chaltikian (BC), Leznov and Blaszak–Marciniak (BM) lattice equations, which are associated with the conformal field theory, UToda $$(m_1,m_2)$$ system and r-matrix, respectively. With symbolic computation, the Bell-polynomial approach is developed to directly bilinearize those three sets of differential–difference nonlinear evolution equations (NLEEs). This Bell-polynomial approach does not rely on any dependent variable transformation, which constitutes the key step and main difficulty of the Hirota bilinear method, and thus has the advantage in the bilinearization of the differential–difference NLEEs. Based on the bilinear forms obtained, the N-soliton solutions are constructed in terms of the $$N \times N$$ Wronskian determinant. Graphic illustrations demonstrate that those solutions, more general than the existing results, permit some new properties, such as the solitonic propagation and interactions for the BC lattice equations, and the nonnegative dark solitons for the BM lattice equations.

Soliton fusion and fission in a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids

Under investigation in this paper is a generalized variable-coefficient fifth-order Korteweg-de Vries equation, which describes the interaction between a water wave and a floating ice cover or the gravity-capillary waves. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear forms, N-soliton solutions, Bäcklund transformation and Lax pair are derived. Infinitely-many conservation laws are obtained based on the Bell-polynomial-typed Bäcklund transformation. Soliton fusion and fission, and influence of the variable coefficients from the equation are analyzed: Both variable coefficients c(t) and n(t) are in direct proportion to the soliton velocities but have no effect on the amplitudes, while another constant coefficient α can affect the types of the interactions, in the sense of the elastic or inelastic. Elastic–inelastic interactions among the three solitons are presented as well.

Bäcklund transformations, Lax system, conservation laws and multisoliton solutions for Jimbo–Miwa equation with Bell-polynomials

Based on binary Bell polynomial approach, the bilinear equation and Ba¨cklund transformations for (3+1)-dimensional Jimbo–Miwa equation are obtained. By virtue of Cole–Hopf transformation, Lax system is constructed by direct linearization of coupled system of binary Bell polynomials. Furthermore, infinite conservation laws are obtained from two field condition in quick and natural way. Finally, a test function of extended three wave method is used to construct multisoliton solutions via bilinear equation.

Bell-Polynomial Approach and Soliton Solutions for Some Higher-Order Korteweg-de Vries Equations in Fluid Mechanics, Plasma Physics and Lattice Dynamics**Supported by the National Natural Science Foundation of China under Grant No. 11272023, the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities

The Korteweg-de Vries (KdV)-type equations have been seen in fluid mechanics, plasma physics and lattice dynamics, etc. This paper will address the bilinearization problem for some higher-order KdV equations. Based on the relationship between the bilinear method and Bell-polynomial scheme, with introducing an auxiliary independent variable, we will present the general bilinear forms. By virtue of the symbolic computation, one- and two-soliton solutions are derived.

Integrability and soliton dynamics for an inhomogeneous nonlinear Schrödinger equation in an inhomogeneous plasma

Under investigation in this paper is an inhomogeneous nonlinear Schrödinger equation, which describes the propagation of a large-wavelength small-amplitude electron plasma wave in a parabolic-distributed and constant-interactional-damping inhomogeneous plasma. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear form, Bäcklund transformation and [Formula: see text]-soliton solutions are obtained. Influence of the linear density coefficient [Formula: see text] and damping coefficient [Formula: see text] on the soliton envelopes is also discussed, i.e. [Formula: see text] can affect the soliton position, while [Formula: see text] is related to the soliton intensity, velocity and phase shift. Periodically attractive and repulsive interactions are shown. Asymptotic analysis shows that the interactions between/among the solitons are elastic.

Soliton excitations and interactions for the three-coupled fourth-order nonlinear Schrödinger equations in the alpha helical proteins

Three-coupled fourth-order nonlinear Schrodinger equations describe the dynamics of alpha helical proteins with the interspine coupling at the higher order. Through symbolic computation and binary Bell-polynomial approach, bilinear forms and N-soliton solutions for such equations are constructed. Key point lies in the introduction of auxiliary functions in the Bell-polynomial expression. Asymptotic analysis is applied to investigate the elastic interaction between the two solitons: two solitons keep their original amplitudes, energies and velocities invariant after the interaction except for the phase shifts. Soliton amplitudes are related to the energy distributed in the solitons of the three spines. Overtaking interaction, head-on interaction and bound-state solitons of two solitons are given. Bound states of three bright solitons arise when all of them propagate in parallel. Elastic interaction between the bound-state solitons and one bright soliton is shown. Increase of the lattice parameter can lead to the increase of the soliton velocity, that is, the interaction period becomes shorter. The solitons propagating along the neighbouring spines are found to interact elastically. Those solitons, exhibited in this paper, might be viewed as a possible carrier of bio-energy transport in the protein molecules.

The Painlevé property, Bäcklund transformation, Lax pair and new analytic solutions of a generalized variable-coefficient KdV equation from fluids and plasmas

A generalized variable-coefficient Korteweg–de Vries (KdV) equation with variable-coefficients of x and t from fluids and plasmas is investigated in this paper. The explicit Painlevé-integrable conditions are given out by Painlevé test, and an auto-Bäcklund transformation is presented via the truncated Painlevé expansion. Under the integrable condition and auto-Bäcklund transformation, the analytic solutions are provided, including the soliton-like, periodic and rational solutions. Lax pair, Riccati-type auto-Bäcklund transformation (R-BT) and Wahlquist–Estabrook-type auto-Bäcklund transformation (WE-BT) are constructed in extended AKNS system. One-soliton-like and two-soliton-like solutions are obtained by R-BT and nonlinear superposition formula is obtained by WE-BT. The bilinear form and N-soliton-like solutions are presented by Bell-polynomial approach. Based on the obtained analytic solutions, the propagation characteristics of waves effected by the variable coefficients are discussed.