Breather Wave Solutions Research Articles

Breather-wave, periodic-wave and traveling-wave solutions for a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid

In this paper, the investigation is conducted on a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid. Via the Riemann theta function, periodic-wave solutions are derived, and breather-wave solutions are constructed with the aid of the extended homoclinic test approach. Based on the polynomial expansion method, several traveling-wave solutions are derived. Besides, we observe that the amplitude of the breather keeps unchanged during the propagation and the traveling wave which is kink shaped propagates stably. Furthermore, we analyze the transition between the periodic-wave and soliton solutions, which implies that the periodic-wave solutions tend to the soliton solutions via a limiting procedure.

Vector breather waves and higher-order rouge waves to the coupled higher-order nonlinear Schrödinger equations

ABSTRACT We investigate the vector breather waves and higher-order rouge waves of the coupled higher-order nonlinear Schrödinger equations via the Darboux-dressing transformation. Firstly, based on the Darboux-dressing transformation, we introduce the asymptotic expansion theory, which allows the solution of the equation to be iterated through the same spectral parameters. Then, the matrix exponential function and the Taylor multi-series expansion method are used to derive the vector breather wave solutions and the higher-order rouge wave solutions. Moreover, we analyze the characteristic lines of the breather waves, which determine the dynamic behaviours of the breather waves. Finally, we give some conclusions of this work.

Dynamics of lump-periodic and breather waves solutions with variable coefficients in liquid with gas bubbles

Lump solutions are empirical rational function solutions found in all directions in space. One of the essential solutions to both linear and nonlinear partial differential equations is lump solutions. The current work studies a class of lump interaction phenomena to the generalized -dimensional nonlinear-wave equation with time-dependent-coefficient. Variable-coefficient nonlinear partial differential equations offer us with more real aspects in the inhomogeneities of media and nonuniformities of boundaries than their counterparts constant-coefficient in many physical cases. The Hirota bilinear form is the fundamental concept that has been used to derive the novel lump-periodic and breather wave solutions. The acquired solutions are constructed using symbolic computations called Maple. The physical characteristics of the acquired solutions are shown with three-dimensional and contour plots in order to shed more light on the acquired novel solutions.

Diverse solitary and Jacobian solutions in a continually laminated fluid with respect to shear flows through the Ostrovsky equation

In this paper, the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods are employed to find the soliton, breather, kink, periodic kink, and lump wave solutions of the Ostrovsky equation. This model is considered as a mathematical modification model of the Korteweg-de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions emerge the localized wave packet as a persistent and dominant feature. Many distinct solutions are obtained through the employed computational schemes. Moreover, some solutions are sketched in 2D, 3D, and contour plots. The effective and powerful of the two used computational schemes are tested. Furthermore, the accuracy of the obtained solutions is examined through a comparison between them and that had been obtained in previously published research.

Darboux-Bäcklund transformation, breather and rogue wave solutions for the discrete Hirota equation

Based on the Lax pair, the pseudopotential and Darboux-Bäcklund transformation of the discrete Hirota equation are given, from which a generalized form of nonlinear wave solutions is derived. By analyzing the properties of the pseudopotential, various types of localized wave solutions including n-direction, t-direction breather wave and rational solutions are obtained and the corresponding dynamical properties and evolutions are given. The obtained results may raise the possibility of related experiments and potential applications in nonlinear science fields, such as nonlinear optics, Bose-Einstein condensates and so on.

Rich analytical solutions of a new (3+1)-dimensional Boiti-Leon- Manna-Pempinelli equation

In this paper, a new (3+1)-dimensional Boiti-Leon- Manna-Pempinelli equation is studied. The interaction solutions of lump and N-soliton (N=2,3,4) are investigated. The interaction solution among lump wave, 1-soliton and periodic wave is presented; The breather-wave solution is obtained. Dynamical behavior is shown by abundant 3D, density and contour plot.

Bilinear Bäcklund transformation, kink and breather-wave solutions for a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics

Fluid mechanics studies are applied in the turbines, airplanes, ships, rivers, windmills, pipes, etc. Under investigation is a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics. Via the Hirota bilinear method, bilinear Backlund transformation is obtained, based on which the kink solutions are constructed. Breather-wave solutions are constructed via the extended homoclinic test approach. When the periods of the breather-wave solutions tend to infinity, the lump solutions are derived. It is found that the amplitudes and shapes of the breather waves keep unchanged during the propagation. Effects of the coefficients in the equation on the amplitudes and velocities for the breather waves are analyzed graphically.

Soliton, breather and rogue wave solutions for the Myrzakulov–Lakshmanan-IV equation

In this paper, we will focus on the Myrzakulov–Lakshmanan-IV (ML-IV) equation. Based on relevant Lax pair, N-fold generalized Darboux Transformation (DT) will be constructed. Some soliton, breather and rogue wave solutions will be derived under different backgrounds through the odtained DT. In the meantime we will analyze the dynamic features of those solutions graphically.

Soliton, breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints* *Project supported by the National Natural Science Foundation of China (Grant No. 11675054), the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213), and the Project of Science and Technology Commission of Shanghai

The nonlinear Schrödinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas. However, due to the difficulty of solving this equation, in particular in high dimensions, lots of methods are proposed to effectively obtain different kinds of solutions, such as neural networks among others. Recently, a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly. Compared with traditional neural networks, this method can obtain remarkably accurate solution with extraordinarily less data. Meanwhile, this method also provides a better physical explanation and generalization. In this paper, based on the above method, we present an improved deep learning method to recover the soliton solutions, breather solution, and rogue wave solutions of the nonlinear Schrödinger equation. In particular, the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time. Moreover, the effects of different numbers of initial points sampled, collocation points sampled, network layers, neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions. Numerical experiments show that the dynamical behaviors of soliton solutions, breather solution, and rogue wave solutions of the integrable nonlinear Schrödinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.

Multiple rogue wave, breather wave and interaction solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation

Based on a direct variable transformation, we obtain multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including first-order, two-order and three-order rogue wave solutions. Their dynamic behaviors are shown by some 3D plots. Compared with Zha’s symbolic computation approach, we do not need to resort to Hirota bilinear form, and it can be used to deal with variable-coefficient integrable equations. Interaction solution between rogue wave and periodic wave is obtained by using the Hirota bilinear form. Abundant breather wave solutions are presented by a direct test function.

Multiple Lump Novel and Accurate Analytical and Numerical Solutions of the Three-Dimensional Potential Yu–Toda–Sasa–Fukuyama Equation

The accuracy of novel lump solutions of the potential form of the three–dimensional potential Yu–Toda–Sasa–Fukuyama (3-Dp-YTSF) equation is investigated. These solutions are obtained by employing the extended simplest equation (ESE) and modified Kudryashov (MKud) schemes to explore its lump and breather wave solutions that characterizes the dynamics of solitons and nonlinear waves in weakly dispersive media, plasma physics, and fluid dynamics. The accuracy of the obtained analytical solutions is investigated through the perspective of numerical and semi-analytical strategies (septic B-spline (SBS) and variational iteration (VI) techniques). Additionally, matching the analytical and numerical solutions is represented along with some distinct types of sketches. The superiority of the MKud is showed as the fourth research paper in our series that has been beginning by Mostafa M. A. Khater and Carlo Cattani with the title “Accuracy of computational schemes”. The functioning of employed schemes appears their effectual and ability to apply to different nonlinear evolution equations.

DARBOUX TRANSFORMATIONS, MULTISOLITONS, BREATHER AND ROGUE WAVE SOLUTIONS FOR A HIGHER-ORDER DISPERSIVE NONLINEAR SCHRÖDINGER EQUATION

<abstract> In this paper, dynamic of a higher-order dispersive nonlinear Schrö-dinger equation is investigated. Firstly, we obtain the determinant representation of the N-fold Darboux transformations of the Schrödinger equation. Then based on the above analysis, we get the one-soliton, two-soliton and the breather wave solution. Furthermore, the first-order rogue wave is derived by means of a Taylor expansion of the breather wave. Finally, by selecting some special parameters and drawing the 3-D and 2-D graphs to better describe the dynamic traits of those solutions. </abstract>

Interaction behaviors between breather and rogue wave in a Heisenberg ferromagnetic equation

A (2+1)-dimensional Heisenberg ferromagnetic equation is under investigation, which is based on Schrödinger equation, and can be used to model nonlinear wave propagation in ferromagnetic spin chain. Analytical mixed breather and rogue wave solution is obtained for the system via the extended Darboux transformation technique. Then the interaction behaviors between breather and rogue wave are studied in details. The nonlinear Kerr effect can influence the wave width, rotation and shape of the mixed waves. Especially, the mixed waves will degenerate into oscillation waves when the Kerr effect is not considered. The breather and rogue wave are squeezed and their amplitudes are amplified by controlling two endogenous parameters.

Bäcklund transformation and localized nonlinear wave solutions of the nonlocal defocusing coupled nonlinear Schrödinger equation

The nonlocal integrable defocusing coupled nonlinear Schrödinger system from a 3×3 spectral problem is investigated. The Bäcklund transformation of the nonlocal defocusing coupled nonlinear Schrödinger system is presented from the pseudopotentials derived from the Lax pair. The nonsingular localized wave solutions including the breather wave and exponential-rational solutions are obtained, whose evolutions and dynamical properties are discussed.

Kinky breather-wave and lump solutions to the (2 + 1)-dimensional Burgers equations

This paper deals with special kinds of exact solutions called kinky breather-wave and lump solutions. In this respect, kinky breather-wave solutions to the (2 + 1)-dimensional Burgers equations are acquired through the use of the extended homoclinic test technique and the Hirota bilinear method. In a special case, bright-dark lump solutions are extracted from kinky breather-wave solutions by adopting some fundamental assumptions. The dynamical behaviors of kinky breather-wave and lump solutions are analyzed by presenting a series of three dimensional and density graphs.

Dynamical analysis of lump, lump-triangular periodic, predictable rogue and breather wave solutions to the (3 + 1)-dimensional gKP–Boussinesq equation

This paper studies the dynamics of shallow water waves with the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq (gKP–Boussinesq) equation arising in fluid mechanics. By the virtue of the Hirota bilinear method, a series of lump, single stripe, lump-triangular periodic, predictable rogue, and breather wave solutions are constructed to the (3 + 1)-D gKP–Boussinesq equation. For generating the breather wave solutions, two homoclinic test approaches are executed to the mentioned equation. It is seen that the breather wave propagates periodically in a certain direction with a constant period. Furthermore, for lump solutions, the shapes and amplitudes of the lump wave are found to be unchanged during its propagation. Besides, the lump-stripe soliton solutions illustrate the fission and fusion behaviors that one-stripe wave splits into one lump and one-stripe waves, and one lump and one-stripe waves merge into one-stripe wave, respectively. To interpret the underlying propagation characteristics, some attained solutions are displayed by making their 3D, density, and contour plots. All the graphs are presented to show their proper wave profiles via the obtained solutions to the studied equation. Moreover, it may be concluded that the attained solutions and their physical features might be helpful to comprehend the localized wave propagation in shallow water waves.

Investigation on breather waves and rogue waves in applied mechanics and physics

Nonlinear evolution equation is a research hot spot in the field of science and engineering. Recently, searching for breather and rogue wave solutions to nonlinear evolution equations has become a popular topic in nonlinear mathematical physics. In this paper, with the help of symbolic calculation, breather and rogue wave solutions to the two generalized (3+1)-dimensional Jimbo-Miwa equation are investigated, respectively. Those analytic solutions are presented, and the influences of the corresponding parameters on breathers and rogue waves are analyzed. Their dynamic properties are discussed based on graphical presentation. Amplitude of breathers and rogue waves are adjusted. Results of this study are helpful to the generation of breather and rogue waves in applied mechanics and physics.

Multi-wave, breather wave and lump solutions of the Boiti–Leon–Manna–Pempinelli equation with variable coefficients

In this paper, a variable-coefficient Boiti–Leon–Manna–Pempinelli equation is to be investigated. We obtain abundant multi-wave, breather wave and lump solutions by using the three waves method, the homoclinic breather approach and the Hirota’s bilinear method. All solutions have been verified to be correct with the help of Mathematica software. We study the propagation characteristics of these solutions by some three-dimensional images. The obtained results are useful for understanding fluid propagating and incompressible fluid.

Breather wave and lump‐type solutions of new (3 + 1)‐dimensional Boiti–Leon–Manna–Pempinelli equation in incompressible fluid

Under investigation is a new (3 + 1)‐dimensional Boiti–Leon–Manna–Pempinelli equation. The main results are listed as follows: (i) lump solutions; (ii) interaction solutions between lump wave and solitary waves; (iii) interaction solutions between lump wave and periodic waves; and (iv) breather wave solutions. Furthermore, graphical representation of all solutions is studied and shown in some 3D and contour plots.

Study on exact solutions of a generalized Calogero–Bogoyavlenskii–Schiff​ equation

By employing the generalized bilinear method and symbolic computation, we obtain new exact solutions of a generalized Calogero–Bogoyavlenskii–Schiff​ equation, which include breather wave solutions, periodic lump soliton solutions, and rational solutions. Via the three-dimensional plot and density plot with the help of Maple, the physical characteristics of these waves are described very well. These outcomes will serve as a very significant milestone in the study of water waves in mathematical physics and engineering phenomena.