Bell-shaped Soliton Solutions Research Articles

Stability analysis and soliton solutions of the (1+1)-dimensional nonlinear chiral Schrödinger equation in nuclear physics

The nonlinear Schrödinger equation equation is one of the most important physical models used in optical fiber theory to explain the transmission of an optical soliton. The field of chiral soliton propagation in nuclear physics is very interesting because of its numerous applications in communications and ultra-fast signal routing systems. The (1+1)-dimensional chiral dynamical structure that describes the soliton behaviour in data transmission is dealt with in this work using a variety of in-depth analytical techniques. This work has applications in particle physics, ionised science, nuclear physics, optics, and other applied mathematical sciences. We are able to develop a variety of solutions to demonstrate the behaviour of solitary wave structures, periodic soliton solutions, chiral soliton solutions, and bell-shaped soliton solutions with the use of applied techniques. Moreover, in order to verify the scientific calculations, the stability analysis for the observed solutions of the governing model is taken into consideration. In addition, the 3-dimensional, contour, and 2-dimensional visuals are supplied for a better understanding of the behaviour of the solutions. The employed strategies are dependable, uncomplicated, and effective; yet have not been utilised with the governing model in the literature that is now accessible. The resulting outcomes have impressive applications across a large number of study areas and computational physics phenomena representing real-world scenarios. The methods applied in this model are not utilized on the given models in previous literature so we can say that these describe the novelty of the work.

The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

Bell-shaped soliton solutions and travelling wave solutions of the fifth-order nonlinear modified Kawahara equation

Abstract The main aim of this work is to investigate numerical solutions of the two different types of the fifth-order modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different type of the Korteweg–de Vries equation. For this approach, we have used an effective and simple type of finite difference method namely Crank-Nicolson scheme for time integration and third-order modified cubic B-spline-based differential quadrature method for space integration. We preferred the third-order modified cubic B-splines to solve the fifth-order partial differential equation because of by using low energy, less algebraic process and produce better results than earlier works. To display the efficiency and accuracy of the present fresh approach famous test problems namely bell-shaped single soliton that has negative amplitude and travelling wave solutions that have the both of the positive and negative amplitudes are solved and the error norms L 2 and L ∞ are calculated and compared with earlier works. Comparison of the error norms show that present fresh approach obtained superior results than earlier works by using same parameters. At the same time, two lowest invariants of the test problems during the simulations are calculated and reported. Besides those, relative changes of invariants are computed and reported.

Highly efficient approach to numerical solutions of two different forms of the modified Kawahara equation via contribution of two effective methods

The numerical solutions of the two different forms of the modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different form of the KdV equation have been investigated. To improve the numerical solutions, two efficient methods have been used together. Firstly, Crank–Nicolson discretization algorithm for time integration is used and then fifth-order quintic B-spline based differential quadrature method for space integration is used. To observe the performance of the present algorithm bell-shaped soliton solution and travelling wave solutions are surveyed. The error norms L2 and L∞ are obtained quite less than earlier papers. The invariants and relative changes of invariants are added to sympathize with superior present results.

Construction of Solitary Two-Wave Solutions for a New Two-Mode Version of the Zakharov-Kuznetsov Equation

A new two-mode version of the generalized Zakharov-Kuznetsov equation is derived using Korsunsky’s method. This dynamical model describes the propagation of two-wave solitons moving simultaneously in the same direction with mutual interaction that depends on an embedded phase-velocity parameter. Three different methods are used to obtain exact bell-shaped soliton solutions and singular soliton solutions to the proposed model. Two-dimensional and three-dimensional plots are also provided to illustrate the interaction dynamics of the obtained two-wave exact solutions upon increasing the phase-velocity parameter.

Lump-Type and Bell-Shaped Soliton Solutions of the Time-Dependent Coefficient Kadomtsev-Petviashvili Equation

In this article, the lump-type solutions of the new integrable time-dependent coefficient (2+1)-dimensional Kadomtsev-Petviashvili equation are investigated by applying the Hirota bilinear technique and a suitable ansatz. The equation is applied in the modelling of propagation of small-amplitude surface waves in large channels or straits of slowly varying width, depth and non-vanishing vorticity. Applying the Bell’s polynomials approach, we successfully acquire the bilinear form of the equation. We firstly find a general form of quadratic function solution of the bilinear form and then expand it as the sums of squares of linear functions satisfying some conditions. Most importantly, we acquire two lump-type and a bell-shaped soliton solutions of the equation. To our knowledge, the lump type solutions of the equation are reported for the first time in this paper. The physical interpretation of the results are discussed and represented graphically.

On the classification of conservation laws and soliton solutions of the long short-wave interaction system

In this paper, the classification of conservation laws (Cls) of the long short-wave interaction system (LSWS) which appears in fluid mechanics as well as plasma physics is implemented using two Cls theorems, namely, the multipliers approach and the new conservation theorem. The LSWS describes the interaction between one long longitudinal wave and one short transverse wave propagating in a generalized elastic medium. The zeroth-order multipliers and the nonlinear self-adjoint substitutions of the model are derived. Considering the fact that the new conservation theorem needs Lie point symmetries in constructing Cls, we derive the point symmetries of a system of nonlinear partial differential equations (NPDEs) acquired by transforming the model into real and imaginary components. Moreover, we derive some kink-type, bell-shaped, singular and combined soliton solutions to the model using the powerful sine-Gordon expansion method (SGEM). Some figures are presented to show the physical interpretations of the acquired results.

Applications of IBSOM and ETEM for solving a discrete electrical lattice

This paper presents a number of new solutions obtained for solving the nonlinear electrical transmission lines by using the improved Bernoulli sub-ODE method and the extended trial equation method. The proposed solutions are kink soliton solution, hyperbolic solution, trigonometric solution and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and concise and can be also applied to other nonlinear partial differential equations. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to construct many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in telecommunication systems where solitons are used for codify data.

Applications of IBSOM and ETEM for solving the nonlinear chains of atoms with long-range interactions

This paper presents a number of new solutions obtained for solving a complex nonlinear equation describing dynamics of nonlinear chains of atoms via the improved Bernoulli sub-ODE method (IBSOM) and the extended trial equation method (ETEM). The proposed solutions are kink solitons, anti-kink solitons, soliton solutions, hyperbolic solutions, trigonometric solutions, and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and succinct and can be also applied to other nonlinear models. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to establish many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in Hamilton’s equations and generalized momentum where solitons are used for long-range interactions.

Nonlinear dynamics of a generalized higher-order nonlinear Schrödinger equation with a periodic external perturbation

The nonlinear dynamics of a generalized higher-order nonlinear Schrodinger (HNLS) equation with a periodic external perturbation is investigated numerically. Via the phase plane analysis, we find that both the homoclinic orbits and heteroclinic orbits can exist for the unperturbed HNLS equation under certain conditions, which respectively corresponds to the bell-shaped and kink-shaped soliton solutions. Moreover, under the effect of the periodic external perturbation, the quasi-periodic bifurcations arise and can evolve into the chaos. The dynamical responses of the perturbed system varying with the perturbation strength and two types of chaotic attractors are discussed to show the existence of the chaotic motions. Via the feedback control methods, such chaotic motions are found to be controlled effectively and finally evolve into the stable quasi-periodic orbits. All the results are helpful to understand the dynamical properties of the nonlinear system.

A Variety of Exact Periodic Wave and Solitary Wave Solutions for the Coupled Higgs Equation

In this work, the coupled Higgs field equation is studied. The extended Jacobi elliptic function expansion methods are efficiently employed to construct the exact periodic solutions of this model. As a result, many exact travelling wave solutions are obtained which include new shock wave solutions or kink-shaped soliton solutions, solitary wave solutions or bell-shaped soliton solutions, and combined solitary wave solutions are formally obtained.

Bell-polynomial approach and soliton solutions for the Zhiber–Shabat equation and (2+1)-dimensional Gardner equation with symbolic computation

Under investigation in this paper are the Zhiber–Shabat and (2+1)-dimensional Gardner equations in quantum fields, fluids and plasmas. Via the Hirota method and symbolic computation, the Bell-polynomial approach is performed to directly bilinearize those equations. For the Zhiber–Shabat equation, based on the bilinear form with an auxiliary variable, the bell-shaped soliton, upside-down bell-shaped soliton and breather-like solutions are obtained. Figures are plotted to illustrate the elastic interactions between two upside-down bell-shaped solitons and the interaction between the breather-like. As to the (2+1)-dimensional Gardner equation, bilinear form, Backlund transformation, one- and two-shock wave solutions are derived. Amplitude-compression and amplification interactions are investigated analytically and graphically.

Solitary Wave and Doubly Periodic Wave Solutions to Three-Dimensional Nizhnik–Novikov–Veselov Equation

The generalized transformation method is utilized to solve three-dimensional Nizhnik–Novikov–Veselov equation and construct a series of new exact solutions including kink-shaped and bell-shaped soliton solutions, trigonometric function solutions, and Jacobi elliptic doubly periodic solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh methods and Jacobi function method, the method we used here gives more general exact solutions without much extra effort.

Explicit exact solutions for a new generalized Hamiltonian amplitude equation with nonlinear terms of any order

Making use of a proper transformation and a generalized ansatz, we consider a new generalized Hamiltonian amplitude equation with nonlinear terms of any order, iu x + u tt + (α|u| p + β|u|2p)u + δu xt = 0. As a result, many explicit exact solutions, which include kink-shaped soliton solutions, bell-shaped soliton solutions, periodic wave solutions, the combined formal solitary wave solutions and rational solutions, are obtained.

Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation**The project supported by National Natural Science Foundation of China under Grant No. 10072013 and the State Key Basic Research Development Program under Grant No. G1998030600

By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.

Extended Hyperbola Function Method and Its Application to Nonlinear Equations**The project supported by National Natural Science Foundation of China under Grant No. 10072013, and the State Key Basic Research Program under Grant No. G1998030600

An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave. The new method can be applied to other nonlinear equations in mathematical physics.

Extended hyperbolic function method and new exact solitary wave solutions of Zakharov equations

With the aid of symbolic computation system Maple and by using the extended hyperbolic function method, the exact solitary wave solutions of Zakharov equations are obtained which include bell-shaped soliton solutions, kink-shaped soliton solutions, envelop soliton solutions, singular soliton solutions and a new type of soliton solutions. The method can be also used to solve other nonlinear wave equations.

A New Algebraic Method for Finding a Series of Travelling Wave Solution to a Coupled Ito System

In this paper a new algebraic method is devised to uniformly construct a series of travelling wave solutions for a coupled Ito system. The obtained solutions include (a) kink-shaped and bell-shaped soliton solutions, (b) rational solutions, (c) triangular periodic solutions, (d) Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic doubly periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives more general exact solutions without much extra effort. More importantly, the method provides a guideline for the classification of the solutions based on the given parameters.

Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method

A new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system. Compared with most of the existing tanh methods, the Jacobi elliptic function method or other sophisticated methods, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters. The solutions obtained in this paper include (a) kink-shaped and bell-shaped soliton solutions, (b) rational solutions, (c) triangular periodic solutions and (d) Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. The efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper, KdV–MKdV, Ito's fifth MKdV, Hirota, Nizhnik–Novikov–Veselov, Broer–Kaup, generalized coupled Hirota–Satsuma, coupled Schrodinger–KdV, (2 + 1)-dimensional dispersive long wave, (2 + 1)-dimensional Davey–Stewartson equations. In addition, as an illustrative sample, the properties of the soliton solutions and Jacobi doubly periodic solutions for the Hirota equation are shown by some figures. The links among our proposed method, the tanh method, extended tanh method and the Jacobi elliptic function method are clarified generally.

Symbolic computation and new families of solitary wave solutions to a Hamiltonian amplitude equation

In this paper, new families of solitary wave solutions are found for a Hamiltonian amplitude equation by using a simple transformation and symbolic computation. These solutions include kink-shaped soliton solutions, bell-shaped soliton solutions and the newly combined kink-shaped and bell-shaped soliton solutions that are of important significance in explaining some physical phenomena.