Rational Wave Solutions Research Articles

New soliton solutions to the perturbed nonlinear Schrödinger equation by exp(− Φ(ξ))-expansion method

The perturbed nonlinear Schrödinger equation (PNLSE), used to investigate the dynamics of wave propagation of light in nonlinear optical fibers and planar wave guides is considered in this article. The model is considered in the presence of full nonlinearity and perturbed terms. By using the state-of-the-art integration scheme, exp(− Φ(ξ))-expansion method, different structures of explicit solutions such as dark, singular, rational and periodic solitary wave solutions are celebrated. All the newly found solutions are discussed with their existence criteria.

Evolutionary behavior of rational wave solutions to the (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation

A nonlinear integrable model known as the (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (4D-BLMP) equation is studied in the present paper. To this end, by considering the Hirota bilinear form of the model and utilizing the linear superposition method (LSM) along with symbolic computations, a group of rational wave solutions including multiple wave and positive (non-singular) compelexiton solutions is formally derived. The dynamical behavior of the solutions is also analyzed graphically by considering the special values of the involved parameters. The results of the current work reveal the existence of different wave structures to the 4D-BLMP equation and distinguish it from other models that do not possess non-singular compelexiton solutions.

High-order lumps, high-order breathers and hybrid solutions for an extended (3 + 1)-dimensional Jimbo–Miwa equation in fluid dynamics

Under investigation in this letter is an extended (3 + 1)-dimensional Jimbo–Miwa (eJM) equation, which can be used to describe many nonlinear phenomena in mathematical physics. With the aid of Hirota bilinear method and long-wave limit method, M-order lumps which describe multiple collisions of lumps are derived. The propagation orbit, velocity and extremum of the 1-order lump solutions on (x, y) plane are investigated in detail. Resorting to the extended homoclinic test technique, we obtain the breather–kink solutions, rational breather solutions and rogue wave solutions for the eJM equation. Meanwhile, through analysis and calculation, the amplitude and period of breather–kink solutions increase with p increasing and the extremum of rational breather solution and rogue waves are also derived. T-order breathers are obtained by means of choosing appropriate complex conjugate parameters on N-soliton solutions. Periods of the 1-order breather solutions on the (x, y) plane are determined by $$k_{12}$$ and $$k_{12}p_{11}+k_{11}p_{12}$$, while locations are determined by $$k_{11}$$ and $$k_{11}p_{11}-k_{12}p_{12}$$. Furthermore, hybrid solutions composed of the kink solitons, breathers and lumps for the eJM equation are worked out. Some figures are given to display the dynamical characteristics of these solutions.

Vector rational and semi-rational rogue waves for the coupled cubic-quintic nonlinear Schrödinger system in a non-Kerr medium

Non-Kerr media possess certain applications in photonic lattices and optical fibers. Studied in this paper are the vector rational and semi-rational rogue waves in a non-Kerr medium, through the coupled cubic-quintic nonlinear Schrödinger system, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium. Applying the gauge transformations, we derive the Nth-order Darboux transformation and Nth-order vector rational and semi-rational rogue wave solutions, where N is a positive integer. With such solutions, we present three types of the second-order rogue waves with the triangle structure: the one with each component containing three four-petaled rogue waves, the one with each component containing three eye-shaped rogue waves, and the other with one component containing three anti-eye-shaped rogue waves and the other component containing three eye-shaped rogue waves. We exhibit the third-order vector rogue waves with the merged, triangle and pentagon structures in each component. Moreover, we show the first- and second-order vector semi-rational rogue waves which display the coexistence of the rogue waves and the breathers.

The new exact solitary solutions for the (3+1)-dimensional Zakharov-Kuznetsov equation using the Riccati equation

In this paper, a non-linear (3+1)-dimensional Zakharov-Kuznetsov equation is investigated by employing the subsidiary equation method, which arises in quantum magneto plasma. The periodic solutions, rational wave solutions, soliton solutions for the quantum Zakharov-Kuznetsov equation which play an important role in mathematical physics are obtained with the help of the Riccati equation expan?sion method. Meanwhile, the electrostatic potential can be accordingly obtained. Compared to the other methods, the exact solutions obtained will extend on earlier reports by using the Riccati equation.

POLYNOMIAL AND RATIONAL WAVE SOLUTIONS OF KUDRYASHOV-SINELSHCHIKOV EQUATION AND NUMERICAL SIMULATIONS FOR ITS DYNAMIC MOTIONS

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions are investigated. Conservation flows of the dynamic motion are obtained utilizing multiplier approach. Using the unified method, a collection of exact solitary and soliton solutions of Kudryashov-Sinelshchikov equation is presented. Collocation finite element method based on quintic B-spline functions is implemented to the equation to evidence the accuracy of the proposed method by test problems. Stability analysis of the numerical scheme is studied by employing von Neumann theory. The obtained analytical and numerical results are in good agreement.

Rational wave solutions to a generalized (2+1)-dimensional Hirota bilinear equation

A generalized form of (2+1)-dimensional Hirota bilinear (2D-HB) equation is considered herein in order to study nonlinear waves in fluids and oceans. The present goal is carried out through adopting the simplified Hirota’s method as well as ansatz approaches to retrieve a bunch of rational wave structures from multiple soliton solutions to breather, rational, and complexiton solutions. Some figures corresponding to a series of rational wave structures are provided, illustrating the dynamics of the obtained solutions. The results of the present paper help to reveal the existence of rational wave structures of different types for the 2D-HB equation.

The solitary waves, breather waves and rogue waves for a generalized nonlinear equation

Under investigation in this work is a generalized nonlinear equation, which can be widely applied to describe various phenomena in nonlinear physical science field. The equation can be reduced to Kadomtsev–Petviashvili-type equations and Jimbo–Miwa-type equations as its special cases. By using Bell’s polynomial, its bilinear representation is well constructed. Based on the obtained bilinear formalism, we derive the solitary wave solutions of the equation. We also consider its kinky breather wave, rational breather wave and rogue wave solutions by employing the extended homoclinic test method, respectively. Moreover, the dynamic behaviors of these solutions are given by graphical analysis. It is hoped that our works can be helpful to understand dynamical behavior of the nonlinear equation.

Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg–de Vries equations

With the aid of a generalized Darboux transformation (DT), we derive a hierarchy of rogue wave solutions to the coupled complex modified Korteweg–de Vries (mKdV) equations. Based on modulation instability (MI), two types of nth order rational rogue wave solutions in compact determinant forms are presented. Especially, the rational rogue wave solutions up to the second-order are performed explicitly and graphically. We find that there exist typical bright–dark composites and rogue wave doublets in first-order case and four or six fundamental rogue waves in second-order case. With appropriate choices of free parameters, the distribution shapes with four fundamental rogue waves admit triangular, quadrilateral, and line structures. Furthermore, triangular, ring, quadrilateral and line patterns can emerge with six fundamental rogue waves. Moreover, we exhibit the first-order semi-rational rogue wave solutions which can demonstrate the coexistence of one rational rogue wave and one breather. Our results can be applicable to the study of rogue wave manifestations in nonlinear optics.

Analytical Wave Solutions for Foam and KdV-Burgers Equations Using Extended Homogeneous Balance Method

In this paper, extended homogeneous balance method is presented with the aid of computer algebraic system Mathematica for deriving new exact traveling wave solutions for the foam drainage equation and the Kowerteg-de Vries–Burgers equation which have many applications in industrial applications and plasma physics. The method is effective to construct a series of analytical solutions including many types like periodical, rational, singular, shock, and soliton wave solutions for a wide class of nonlinear evolution equations in mathematical physics and engineering sciences.

General rogue wave solutions of the coupled Fokas-Lenells equations and non-recursive Darboux transformation.

We formulate a non-recursive Darboux transformation technique to obtain the general nth-order rational rogue wave solutions to the coupled Fokas-Lenells system, which is an integrable extension of the noted Manakov system, by considering both the double-root and triple-root situations of the spectral characteristic equation. Based on the explicit fundamental and second-order rogue wave solutions, we demonstrate several interesting rogue wave dynamics, among which are coexisting rogue waves and anomalous Peregrine solitons. Our solutions are generalized to include the complete background-field parameters and therefore helpful for future experimental study.

Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity

Abstract In this work, we consider the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions are obtained by means of the unified method. The solutions showed that this method provides us with a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences.

Darboux Transformations, Higher-Order Rational Solitons and Rogue Wave Solutions for a (2 + 1)-Dimensional Nonlinear Schrödinger Equation**Supported by National Natural Science Foundation of China under Grant Nos. 11775121 and 11435005, and K.C. Wong Magna Fund in Ningbo University

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations (DTs) for a (2 + 1)-dimensional nonlinear Schrödinger (NLS) equation is derived, which can be reduced to two (1 + 1)-dimensional equation: a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave (RW) solutions are constructed by its (1, N −1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.

New solitary wave solutions of (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation

In this article, via the improved fractional subequation method, the fully analytical solutions of the (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation involving Jumarie’s modified Riemann–Liouville derivative have been derived. As a result, with the aid of symbolic computation, many types of new analytical solutions are obtained, which include new solitary wave, periodic wave and rational wave solutions. The graphical representations show that these gained solutions have abundant structures.

Resonant multiple wave solutions, complexiton solutions and rogue waves of a generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles

ABSTRACT In this paper, the generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles is investigated. It describes the dynamics of nonlinear waves in hydrodynamics. On the one hand, we construct a class of special N-wave solutions by applying the linear superposition principle, resonant multiple wave solutions and complexiton solutions are derived. On the other hand, the rational breather wave and rogue wave solutions of the equation are also obtained by employing the extended homoclinic test method. Finally, all these solutions are presented via 3-dimensional plots and density plots with choices some special parameters to show the dynamic characteristics.

Rogue waves, bright–dark solitons and traveling wave solutions of the [formula omitted]-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, a (3+1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation is investigated, which describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. By employing the extended homoclinic test method, we construct a new family of two wave solutions, rational breather wave and rogue wave solutions of the equation. Moreover, by virtue of some ansatz functions and the Riccati equation method, its analytical bright soliton, dark soliton and traveling wave solutions are derived. Finally, we obtain its exact power series solution with the convergence analysis. In order to further understand the dynamics, we provide some graphical analysis of these solutions.

Three-component nonlinear Schrödinger equations: Modulational instability, Nth-order vector rational and semi-rational rogue waves, and dynamics

The integrable three-component nonlinear Schrödinger equations are systemically explored in this paper. We firstly find the conditions for the modulational instability of plane-wave solutions of the system. Secondly, we present the general formulae for the Nth-order vector rational and semi-rational rogue wave solutions by the generalized Darboux transformation and formal series method. Particularly, we find that the second-order vector rational RWs contain five, seven, and nine fundamental vector RWs, which can arrange with many novel excitation dynamical patterns such as pentagon, triangle, ‘clawlike’, line, hexagon, arrow, and trapezoid structures. Moreover, we also find two different kinds of Nth-order vector semi-rational RWs: one of which can demonstrate the coexistence of Nth-order vector rational RW and N parallel vector breathers and the other can demonstrate the coexistence of Nth-order vector rational RWs and Nth-order Y-shaped vector breathers. We also exhibit distribution patterns of superposition of RWs, which can be constituted of different fundamental RW patterns. Finally, we numerically explore the dynamical behaviors of some chosen RWs. The results could excite the interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids.

On nonautonomous complex wave solutions described by the coupled Schrödinger–Boussinesq equation with variable-coefficients

This paper investigates the coupled Schrodinger–Boussinesq equation with variable-coefficients using the unified method. New nonautonomous complex wave solutions are obtained and classified into two categories, namely polynomial function and rational function solutions. For the polynomial functions emerge the complex solitary, complex soliton and complex elliptic wave solutions, while for the rational function are observed complex periodic rational and complex hyperbolic rational wave solutions. The physical insight and the dynamical behavior of the solutions describing the wave propagation in laser or plasma physics are discussed and analysed for different choices of the arbitrary functions in the solutions.

Interaction solutions of a (2[formula omitted]1)-dimensional dispersive long wave system

With the help of the consistent tanh expansion, this paper obtains the interaction solutions between solitons and potential Burgers waves of a (2+1)-dimensional dispersive long wave system. Based on some known solutions of the potential Burgers equation, the multiple resonant soliton wave solutions, soliton–error function wave solutions, soliton–rational function wave solutions and soliton–periodic wave solutions are obtained directly.

Solitary waves, homoclinic breather waves and rogue waves of the [formula omitted]-dimensional Hirota bilinear equation

In this paper, the (3+1)-dimensional Hirota bilinear equation is investigated, which can be used to describe the nonlinear dynamic behavior in physics. By using the Bell polynomials, the bilinear form of the equation is derived in a very natural way. Based on the resulting bilinear form, its N-solitary waves are further obtained by using the Hirota’s bilinear theory. Finally, by using the Homoclinic test method, we obtain its rational breather wave and rogue wave solutions, respectively. In order to better understand the dynamical behaviors of the equation, some graphical analyses are discussed for these exact solutions.