Abundant Exact Traveling Wave Solutions Research Articles

Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics.

Diverse and novel soliton structures of coupled nonlinear Schrödinger type equations through two competent techniques

Nonlinear evolution equations play enormous significant roles to work with complicated physical phenomena located across the nature world. The Schrödinger type equations bearing nonlinearity are important models that flourished with the wide-ranging arena concerning plasma physics, nonlinear optics, fluid flow and the theory of deep-water waves. In this exploration, we retrieve the soliton and other solutions in an appropriate form to the coupled nonlinear Schrödinger equations by means of the improved tanh method and the rational [Formula: see text]-expansion method. The suggested system of nonlinear Schrödinger equations is turned into a differential equation of a single variable through executing some operations. Thereupon, successful implementation of the advised techniques regains the abundant exact traveling wave solutions. The obtained solutions are figured out in the profiles of three-dimensional (3D), two-dimensional (2D) and contour by assigning suitable values of the involved unknown constants. These diverse graphical appearances enable the researchers to understand the underlying mechanisms of intricate phenomena of the leading equations. The individual performances of the employed methods are praiseworthy which deserve further application to unravel any other nonlinear partial differential equations arising in various branches of science.

On new abundant exact traveling wave solutions to the local fractional Gardner equation defined on Cantor sets

In this article, we derive a new local fractional Gardner equation based on the local fractional derivative. A novel traveling wave transform of the non‐differentiable type is utilized to convert the local fractional Gardner equation into a nonlinear local fractional ordinary differential equation. Then a new method called the Mittag–Leffler function based method is proposed for the first time ever to construct the abundant traveling wave solutions. By this method, three families (six sets) of the traveling wave solutions on Cantor set are obtained. Finally, the performances of the various solutions on Cantor set are presented through the three‐dimensional contours. It shows that the proposed method is straightforward, powerful and can provide more forms of the traveling wave solutions for the local fractional equations, which is expected to be helpful for the study of the traveling wave theory for local fractional equations.

Analytical treatment in optical metamaterials with anti-cubic law of nonlinearity by IEFM and EShGEEM

Here, the improved exp(−Ω())-expansion method (IEFM) and extended sinh-Gordon equation expansion method (EShGEEM)are being applied on (1+2)-dimensional nonlinear Schr¨odinger equation, optical metamaterials, with anti-cubic nonlinearity. Materialslike photovoltaic-photorefractive, polymer and organic consists of spatial solitons and optical nonlinearities, which can beidentified by seeking help from NLSE with anti-cubic nonlinearity. Abundant exact traveling wave solutions consisting of freeparameters are established in terms of bright, dark, singular, kink-singular, and combined dark-bright soliton solutions. Variousarbitrary constants obtained in the solutions help us to discuss the graphical behavior of solutions and also grants flexibilityto formulate solutions that can be linked with a large variety of physical phenomena. Moreover, graphical representation ofsolutions are shown vigorously in order to visualize the behavior of the solutions acquired for the equation.

Solitary Wave Solutions for $$(1+2)$$-Dimensional Nonlinear Schrödinger Equation with Dual Power Law Nonlinearity

Here, \(\tan \left( \frac{\phi (\xi )}{2}\right) \)-expansion method is being applied on \((1+2)\)-dimensional nonlinear Schrodinger equation (NLSE) with dual power law nonlinearity. Spatial solitons and optial nonlinearities in materials like photovoltaic–photorefractive, polymer and organic can be identified by seeking help from NLSE with dual power law nonlinearity. Abundant exact traveling wave solutions consisting free parameters are established in terms of exponential functions. Various arbitrary constants obtained in the solutions help us to discuss the graphical behavior of solutions and also grants flexibility to form a link with large variety of physical phenomena. Moreover, graphical representation of solutions are shown vigorously in order to visualize the behavior of the solutions acquired for the equation.

Novel approaches to extract soliton solutions of the (1 + 1) dimensional Fokas-Lenells equation by means of the complex transformation

This article presents the complex transformation to extract the traveling wave solutions of the Fokas-Lenells equation using the uniform algebraic approach and extended G′G2-expansion approach. These methods are direct techniques to find the exact traveling wave solutions. Abundant exact traveling wave solutions are obtained by uniform algebraic method and extended G′G2-expansion method which are expressed by the solitary, periodic, trigonometric functions, hyperbolic functions and rational functions. The graphical representation is given to visualize the nature of the obtained solutions by selecting the particular values of the parameters.

Exact Solutions to Time-Fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry

We study a fifth order time-fractional KdV equation (FKdV) under meaning of the conformal fractional derivative. By trial equation method based on symmetry, we construct the abundant exact traveling wave solutions to the FKdV equation. These solutions show rich evolution patterns including solitons, rational singular solutions, periodic and double periodic solutions and so forth. In particular, under the concrete parameters, we give the representations of all these solutions.

Analytical study of resonant optical solitons with variable coefficients in Kerr and non-Kerr law media

In this article, the extended trial approach has been used for the investigation of traveling wave solutions of resonant nonlinear Schrodinger equation with variable coefficients depending upon time in the presence of different effects such as Kerr law and parabolic law nonlinearity including a Bohm potential term. Many new soliton solutions such as bright, dark, singular and rational function solutions are obtained. This technique proves to be an advantageous and powerful mathematical tool for establishing abundant exact traveling wave solutions in mathematical physics and plasma. The resonant nonlinear Schrodinger’s equation, in specific, governs the propagation of soliton in both quantum potential and uniaxial waves in a cold collisionless plasma.

Applications of He's semi-inverse method, ITEM and GGM to the Davey-Stewartson equation

We investigate the Davey-Stewartson (DS) equation. Travelling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, He's semi-inverse variational principle method (SIVPM), the improved \(\tan(\phi/2)\)-expansion method (ITEM) and generalized \(G'/G\)-expansion method (GGM) for seeking more exact solutions via the DS equation. These methods are direct, concise and simple to implement compared to other existing methods. The exact solutions containing four types solutions have been achieved. The results demonstrate that the aforementioned methods are more efficient than the Ansatz method applied by Mirzazadeh (2015). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found by the improved \(\tan(\phi/2)\)-expansion and generalized \(G'/G\)-expansion methods. By He's semi-inverse variational principle we have obtained dark and bright soliton wave solutions. Also, the obtained semi-inverse variational principle has profound implications in physical understandings. These solutions might play important role in engineering and physics fields. Moreover, by using Matlab, some graphical simulations were done to see the behavior of these solutions.

Comparison between the generalized tanh–coth and the (G′/G)-expansion methods for solving NPDEs and NODEs

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.

The exact solutions and approximate analytic solutions of the (2+1)-dimensional KP equation based on symmetry method.

In this paper, we successfully obtained the exact solutions and the approximate analytic solutions of the (2 + 1)-dimensional KP equation based on the Lie symmetry, the extended tanh method and the homotopy perturbation method. In first part, we obtained the symmetries of the (2 + 1)-dimensional KP equation based on the Wu-differential characteristic set algorithm and reduced it. In the second part, we constructed the abundant exact travelling wave solutions by using the extended tanh method. These solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively. It should be noted that when the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions. Finally, we apply the homotopy perturbation method to obtain the approximate analytic solutions based on four kinds of initial conditions.

Analytic study of sixth-order thin-film equation by tan(ϕ/2)-expansion method

A improvement of the expansion methods, namely, the improved $$\tan (\phi (\xi )/2)$$ -expansion method for solving the sixth-order thin-film equation is proposed. As a result, many new and more general exact traveling wave solutions are obtained including singular kink-type solutions. We obtained the further solutions comparing with other methods as Flitton and King (Eur J Appl Math 15:713–754, 2004) and Taha et al. (J King Saud Univ Sci 26:75–78, 2014). Recently this method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including kink and rational solutions have been found. These solutions might play important role in engineering and physics fields. Also the results demonstrate that the introduced method is powerful tools for solving the nonlinear partial differential equations.

Optical soliton wave solutions to the resonant Davey–Stewartson system

We investigate the resonant Davey–Stewartson (DS) system. The resonant DS system is a natural \((2+1)\)-dimensional version of the resonant nonlinear Schrodinger equation. Traveling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, improved \(\tan (\phi /2)\)-expansion method (ITEM) and generalized (G′/G)-expansion method for seeking more exact solutions via the resonant Davey–Stewartson system. These methods are direct, concise and simple to implement compared to other existing methods. The exact particular solutions containing four types of solutions, i.e., hyperbolic function, trigonometric function, exponential and solutions. We obtained further solutions comparing these methods with other methods. The results demonstrate that the aforementioned methods are more efficient than the multi-linear variable separation method applied by Tang et al. (Chaos Solitons Fractals 42:2707–2712, 2009). Recently the ITEM was developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering and physics fields. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving the nonlinear problems.

Abundant soliton solutions for the Kundu–Eckhaus equation via tan(ϕ(ξ))-expansion method

In this paper, the improved tanΦ(ξ)/2-expansion method is proposed to seek more general exact solutions of the Kundu–Eckhaus equation. Being concise and straightforward, this method is applied to the nonlinear Kundu–Eckhaus equation. The exact particular solutions containing five types hyperbolic function solution (exact soliton wave solution), trigonometric function solution (exact periodic wave solution), rational exponential solution (exact singular kink-type wave solution), logarithmic solution and rational solution (exact singular cupson wave solution). We obtained further solutions comparing this method with other methods. The results demonstrate that the new tanΦ(ξ)/2-expansion method is more efficient than the Ansatz method applied by Baskonus et al. [16]. Recently this method developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering and physics fields.

Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics

A improvement of the expansion methods namely the improved $$\tan \left( \varPhi (\xi )/2\right)$$ -expansion method for solving the Tzitzeica type nonlinear evolution equations is proposed. In this work, the dispersive optical solitons that are governed by the Tzitzeica type nonlinear evolution equations. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions and trigonometric function solutions. The exact particular solutions containing four types hyperbolic function solution, trigonometric function solution, exponential solution and rational solution. We obtained the further solutions comparing with other methods. Recently this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear problems.

New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients

An improvement of the expansion methods, namely, the improved tan⁡Φξ/2-expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions, and trigonometric function solutions. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions comparing this method with other methods. The results demonstrate that the new tan⁡Φξ/2-expansion method is more efficient than the Ansatz method and Tanh method applied by Triki and Wazwaz (2013). Recently, this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions have been found. These solutions might play an important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear physics.

Application of tan(ϕ/2)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity

In this paper, the improved tan(Φ(ξ)/2)-expansion method is proposed to seek more general exact solutions of the new partial differential equation. Being concise and straightforward, this method is applied to the Biswas–Milovic equation (BME). The exact particular solutions containing four types hyperbolic function solution, trigonometric function solution, exponential solution and rational solution. We obtained the further solutions comparing with other methods as [13]. Recently this method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering and physics fields. It is shown that this method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving problems in nonlinear optic.

NEW TRAVELLING WAVE SOLUTIONS FOR STOCHASTIC CUBIC NON LINEAR KLEIN-GORDON EQUATION

This paper is devoted to give white noise functional solutions for stochastic cubic nonlinear Klein-Gordon equation. Abundant exact travelling wave solutions for Wick-type generalized stochastic Klein-Gordon equation are obtained. By using white noise analysis, Hermite transform, modified Riccati equation and modified tanh-coth method many exact travelling wave solutions are given. Detailed computations and implemented examples for the investi- gated model are explicitly provided.

Modified alternative (G′/G)-expansion method to general Sawada–Kotera equation of fifth-order

Abstract The alternative ( G ′ / G ) -expansion method has been further modified by introducing the generalized Riccati equation to construct new exact solutions. In order to illustrate the novelty and advantages of this approach, the general Sawada–Kotera (GSK) equation is considered and abundant new exact traveling wave solutions are obtained in a uniform way. These solutions may be imperative and significant for the explanation of some practical physical phenomena. It is shown that the modified alternative ( G ′ / G ) -expansion method is an efficient and advance mathematical tool for solving nonlinear partial differential equations in mathematical physics.

Exact Traveling Wave Solutions for Wick-Type Stochastic Schamel KdV Equation

F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. By means of Hermite transform, inverse Hermite transform, and white noise analysis, the variable coefficients and Wick-type stochastic Schamel KdV equations are completely described. Abundant exact traveling wave solutions for variable coefficients Schamel KdV equations are given. These solutions include exact stochastic Jacobi elliptic functions, trigonometric functions, and hyperbolic functions solutions.