Algebraic Solitary Wave Solution Research Articles

The Perturbed Chen–Lee–Liu Equation: Diverse Optical Soliton Solutions and Other Wave Solutions

The objective of the present study is to extract the optical soliton solutions (OSSs) of the perturbed Chen–Lee–Liu equation by exerting three techniques, which are the extended Wang’s direct mapping method, tanh/coth function method and the Subequation method. Various OSSs, including the dark, bright, and singular soliton solutions, as well as other wave solutions, such as the singular periodic and algebraic solitary wave solutions, are successfully found. By assigning the appropriate parameters, the dynamic characteristics of the attained solutions are depicted graphically via the Mathematica software. The outcomes confirm that the methods are highly effective for constructing the OSSs and can also be adopted to investigate the other PDEs in optics.

Diverse optical solitons to the Radhakrishnan–Kundu–Lakshmanan equation for the light pulses

In this paper, we study the Radhakrishnan–Kundu–Lakshmanan equation (RKLE) that plays a key role in modeling the nonlinear propagation of the light pulses. Two novel methods namely Wang’s Bäcklund transformation-based method and Wang’s direct mapping method are employed to construct the diverse exact optical soliton solutions. Abundant exact solutions like the bright soliton solution, dark soliton solutions, singular soliton solutions, algebraic solitary wave solutions and the singular periodic wave solutions expressed in the form of the rational function type, double-exp function type, Sin-Cos function type, Tan function type, Cot function type, Sinh-Cosh function type, Tanh function type, Coth function type, Sech function type, Csch function type, Sec function type and Csc function type are obtained. The dynamics of the different exact solutions are presented through the 3D plot, contour and 2D curve, and the corresponding physical interpretations are elaborated in detail. It is confirmed that the proposed methods are effective and powerful, and can be adopted to construct the optical solitons of the other PDEs arising in the optical physics.

BÄCKLUND TRANSFORMATION AND DIVERSE EXACT EXPLICIT SOLUTIONS OF THE FRACTAL COMBINED KdV–mKdV EQUATION

A fractal modification of the combined KdV–mKdV equation which plays a key role in various fields of physics is presented in this work for the first time. Aided by the fractal two-scale transform, the homogeneous balance method is employed to construct the fractal Bäcklund transformation. By means of the Bäcklund transformation, some new exact explicit solutions such as the algebraic solitary wave solution of rational function, single-soliton solution, double-soliton solutions, N-soliton solutions, singular traveling solutions and the periodic wave solutions of trigonometric function are obtained. Finally, some solutions are illustrated with different fractal orders in the form of the 3D plot, 3D density and 2D curves by assigning reasonable parameters with the help of Mathematica. The findings in this paper are expected to present some new insights into the fractal theory of the fractal PDEs.

Chirped self-similar solitary waves for the generalized nonlinear Schrödinger equation with distributed two-power-law nonlinearities.

We investigate the propagation characteristics of the chirped self-similar solitary waves in non-Kerr nonlinear media within the framework of the generalized nonlinear Schrödinger equation with distributed dispersion, two-power-law nonlinearities, and gain or loss. This model contains many special types of nonlinear equations that appear in various branches of contemporary physics. We extend the self-similar analysis presented for searching chirped self-similar structures of the cubic model to a more general problem involving two nonlinear terms of arbitrary power. A variety of exact linearly chirped localized solutions with interesting properties are derived in the presence of all physical effects. The solutions comprise bright, kink and antikink, and algebraic solitary wave solutions, illustrating the potentially rich set of self-similar pulses of the model. It is shown that these optical pulses possess a linear chirp that leads to efficient compression or amplification, and thus are particularly useful in the design of optical fiber amplifiers, optical pulse compressors, and solitary wave based communication links. Finally, the stability of the self-similar solutions is discussed numerically under finite initial perturbations.

The Klein–Gordon–Zakharov equations with the positive fractional power terms and their exact solutions

In this paper, the famous Klein-Gordon-Zakharov equations are firstly generalized, the new special types of Klein-Gordon-Zakharov equations with the positive fractional power terms (gKGZE) are presented. In order to derive the exact solutions of new special gKGZE, the subsidiary higher order ordinary differential equations (sub-ODEs) with the positive fractional power terms are introduced, and with the aids of the Sub-ODE, the exact solutions of three special types of the gKGZE are derived, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of gKGZE satisfy certain constraint conditions.

Interaction Between Kink Solitary Wave and Rogue Wave for (2+1)-Dimensional Burgers Equation

Based on a suitable ansatz approach and Hirota’s bilinear form, kink solitary wave, rogue wave and mixed exponential–algebraic solitary wave solutions of (2+1)-dimensional Burgers equation are derived. The completely non-elastic interaction between kink solitary wave and rogue wave for the (2+1)-dimensional Burgers equation are presented. These results enrich the variety of the dynamics of higher dimensional nonlinear wave field.

Bäcklund transformations and abundant exact explicit solutions of the Sharma–Tasso–Olver equation

By using an extension of the homogeneous balance method and Maple, the Bäcklund transformations for the Sharma–Tasso–Olver equation are derived. The connections between the Sharma–Tasso–Olver equation and some linear partial differential equations are found. With the aid of the transformations given here and the computer program Maple 12, abundant exact explicit special solutions to the Sharma–Tasso–Olver equation are constructed. In addition to all known solutions re-deriving in a systematic way, several entirely new and more general exact explicit solitary wave solutions can also be obtained. These solutions include (a) the algebraic solitary wave solution of rational function, (b) single-soliton solutions, (c) double-soliton solutions, (d) N-soliton solutions, (e) singular traveling solutions, (f) the periodic wave solutions of trigonometric function type, and (g) many non-traveling solutions. By using the Airy’s function and the Bäcklund transformations obtained here, the exact explicit solution of the initial value problem for the STO equation is presented. The variety of the structure of the solutions for the Sharma–Tasso–Olver equation is illustrated.

Periodic Wave Solutions of a Generalized KdV-mKdV Equation with Higher-Order Nonlinear Terms

The Jacobin doubly periodic wave solution, the Weierstrass elliptic function solution, the bell-type solitary wave solution, the kink-type solitary wave solution, the algebraic solitary wave solution, and the triangular solution of a generalized Korteweg-de Vries-modified Korteweg-de Vries equation (GKdV-mKdV) with higher-order nonlinear terms are obtained by a generalized subsidiary ordinary differential equation method (Gsub-ODE method for short). The key ideas of the Gsub-ODE method are that the periodic wave solutions of a complicated nonlinear wave equation can be constructed by means of the solutions of some simple and solvable ODE which are called Gsub-ODE with higherorder nonlinear terms

Solitary Wave Solutions of a Generalized Derivative Nonlinear Schrödinger Equation

With the aid of a class of nonlinear ordinary differential equation (ODE) and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Schrödinger equation (GDNLSE) have been found out, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of GDNLSE satisfy certain constraint conditions. For more general GDNLSE, the similar results are also given.

A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms

The bell type solitary wave solution, the kink type solitary wave solution, the algebraic solitary wave solution and the sinusoidal traveling wave solution of a generalized KdV–mKdV equation (GKdV–mKdV) with high-order nonlinear terms are obtained by a subsidiary ordinary differential equation method (sub-ODE method for short). The key ideas of the sub-ODE method are that the traveling wave solutions of a complicated nonlinear wave equation can be constructed by means of the solutions of some simple and solvable ODEs which are called sub-ODEs.

Existence of solitary internal waves in a two-layer fluid of infinite depth

Abstract : This paper concerns the existence of internal solitary waves moving with a constant speed at the interface of a two-layer fluid with infinite height. The fluids are immiscible, inviscid, and incompressible with constant but different densities. Assume that the height of the upper fluid is infinite and the depth of the lower fluid is finite. It has been formally derived before that under long-wave assumption the first-order approximation of the interface satisfies the Benjamin-Ono equation, which has algebraic solitary-wave solutions. This paper gives a rigorous proof of the existence of solitary-wave solutions of the exact equations governing the fluid motion, whose first-order approximations are the algebraic solitary-wave solutions of the Benjamin-Ono equation. The proof relies on estimates of integral operators using Fourier transforms in L2(R)- space and is different from the previous existence proof of solitary waves in a two-layer fluid with finite depth.

Stability of the Bright-Type Algebraic Solitary-Wave Solutions of Two Extended Versions of the Nonlinear Schrödinger Equation

The stability of the bright-type algebraic solitary-wave solutions of two extended nonlinear Schrodinger (NLS) equations, recently obtained by Hayata and Koshiba (HK) [Phys. Rev. E 51 (1995) 1499], is investigated by means of the averaged Lagrangian variational technique. Concerning the first equation (a quadratic-cubic NLS eq.), the variational analysis shows that the exact solution found by HK is stable, as solitary-wave initial conditions which deviate from the HK solution evolve into oscillatory functions whose envelopes remain close to the exact solution, if certain stability conditions are satisfied. Concerning the second equation (a cubic-quintic NLS eq.), the variational analysis indicates that in this case the exact solution is unstable, as solitary-wave initial conditions which are higher than the exact solution lead to a blowup within a finite distance, whereas initial pulses which are lower than the exact solution are completely dispersed.

Algebraic solitary-wave solutions of a nonlinear Schrödinger equation.

We present algebraic solitary-wave solutions of a nonlinear Schroedinger equation that includes two terms with power-law nonlinearity. Both bright and dark types are found. Numerical stability analyses show that the solitary wave exhibits a solitonic feature.

Algebraic rossby solitary wave and blocking in the atmosphere

In this paper, the atmosphere is divided into two regions which are governed by different equations. Furthermore, multiple-scale method is used to obtain the Benjamin-Ono equation satisfied by the nonlinear Rossby wave in weak shear zonal flow. The equation has algebraic solitary wave solution, and the stationary streamfunction fields in-ithe atmosphere are calculated by using numerical method, and the results demonstrate that the stationary solution is antisymmetric dipoles with the anticyclone north of the cyclone, and the structure is similar to the equivalent modon obtained by Mcwilliams (1980). Because the modon obtained here is an algebraic solitary wave, the modon may be called “algebraic modon”, and the stationary algebraic modon is consistent with observations of blocking pattern in the atmosphere.

Periodic and aperiodic solitary wave solutions of the nonlinear Klein-Gordon equation without dispersion

The Klein-Gordon equation without dispersion, and with quadratic and cubic nonlinearities, has been studied in one and higher dimensions. Algebraic solitary wave solutions in all cases, as well as higher-order modes in higher dimensions (similar to nonlinear optics) have been shown to exist corresponding to specific initial values. While in the one-dimensional case, arbitrary initial values yield periodic solutions, asymptotically stable solutions are shown to exist in the higher-dimensional case. For both one- and higher-dimensional cases, solutions tending to zero with distance are shown to be achieved for other initial conditions by incorporating a small amount of 'saturating' fourth-order nonlinearity. Finally, it is shown how a general Klein-Gordon equation with dispersion and a forcing term may be reduced to the equation discussed in the paper.