Explicit Solitary Wave Solutions Research Articles

Bifurcations and Exact Solutions of Optical Soliton Models in Fifth-Order Weakly Nonlocal Nonlinear Media

For the optical soliton model in fifth-order weakly nonlocal nonlinear media, to find its exact explicit solutions, the corresponding traveling wave system is formulated as a planar dynamical system with a singular straight line. Then, by using techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze the planar system and find the corresponding phase portraits, the dynamical behavior of the amplitude component can be assessed. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink, and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons are found with precise formulations.

Exploring analytical results for (2+1) dimensional breaking soliton equation and stochastic fractional Broer-Kaup system

<abstract> <p>This paper introduces a pioneering exploration of the stochastic (2+1) dimensional breaking soliton equation (SBSE) and the stochastic fractional Broer-Kaup system (SFBK), employing the first integral method to uncover explicit solutions, including trigonometric, exponential, hyperbolic, and solitary wave solutions. Despite the extensive application of the Broer-Kaup model in tsunami wave analysis and plasma physics, existing literature has largely overlooked the complexity introduced by stochastic elements and fractional dimensions. Our study fills this critical gap by extending the traditional Broer-Kaup equations through the lens of stochastic forces, thereby offering a more comprehensive framework for analyzing hydrodynamic wave models. The novelty of our approach lies in the detailed investigation of the SBSE and SFBK equations, providing new insights into the behavior of shallow water waves under the influence of randomness. This work not only advances theoretical understanding but also enhances practical analysis capabilities by illustrating the effects of noise on wave propagation. Utilizing MATLAB for visual representation, we demonstrate the efficiency and flexibility of our method in addressing these sophisticated physical processes. The analytical solutions derived here mark a significant departure from previous findings, contributing novel perspectives to the field and paving the way for future research into complex wave dynamics.</p> </abstract>

Explicit solitary wave solutions for the nonlinear equations in semiconductor and magnetic field with their stability analysis

Explicit solitary wave solutions for the nonlinear equations in semiconductor and magnetic field with their stability analysis

The plethora of exact and explicit soliton solutions of the hyperbolic local (4+1)-dimensional BLMP model via GERF method

In this article, some novel results to the hyperbolic local (4+1)-dim Boiti-Leon-Manna-Pempinelli (BLMP) equation signifies the wave propagation over an incompressible fluid. In comparison to deep water, more essential wave capacity conditions are required for the linearization of the wave structure and strong nonlinear aspects are evident in shallow water. The explicit solitary wave solutions of the hyperbolic local (4+1)-dim BLMP equation are secured in this study using the generalized exponential rational function (GERF) approach. The resulting solutions are also validated by the use of computational tools like Maple. The dynamic characteristics of selected obtained solutions are depicted in 3D along their projections and as 2D graphs for various choices of arbitrary parameters. The outcomes demonstrate the effectiveness of the GERF approach for determining the sws of the (4+1)-dim BLMP equation.

Head-on collision of magnetosonic solitary waves at low latitudes ionosphere plasma

The dynamics of head-on collision of two magnetosonic solitary waves in ionosphere plasma of the Earth is investigated. A geomagnetic field model standing for the relation between a magnetic field and latitude is used to study nonlinear magnetosonic waves in the ionosphere of Earth. The linear and weakly nonlinear properties of magnetosonic waves are studied by the dispersion relation and the extended Poincaré–Lighthill–Kuo (PLK) method, respectively. Two coupled damped Korteweg–de Vries equations (dKdV) are derived for oppositely propagating magnetosonic solitary waves. The explicit solitary wave solutions are obtained in the weak collision limit, and the trajectories and phase shifts of two magnetosonic solitary waves are derived, which show that collisional dynamics and their phase shifts are dependent on the collision caused by neutral particles and the properties of the ionosphere. This study is applied to investigate the two-counterpropagating magnetosonic solitary waves located in the F2-region of the Earth's ionosphere at low latitude. This study should be beneficial to understand the interaction dynamics of the head-on nonlinear magnetosonic waves located in the F2 layer of the ionosphere when collision effects caused by neutral particles and geomagnetic field distribution are considered.

Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel–Korteweg–de Vries equation* *Project supported by the National Natural Science Foundation of China (Grant No. 11461022).

The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics. The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses. The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given. All possible exact explicit parametric representations of the waves are also presented. Along with the details of the analyses, the analytical results are numerically simulated lastly.

BIFURCATIONS AND EXACT TRAVELLING WAVE SOLUTIONS OF M-N-WANG EQUATION

By using the method of dynamical systems to Mikhailov-Novikov-Wang Equation, through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system of the derivative $\phi(\xi)$ of the wave function $\psi(\xi)$. Under different parameter conditions, for $\phi(\xi)$, exact explicit solitary wave solutions, periodic peakon and anti-peakon solutions are obtained. By integrating known $\phi(\xi)$, nine exact explicit traveling wave solutions of $\psi(\xi)$ are given.

Rational Waves and Complex Dynamics: Analytical Insights into a Generalized Nonlinear Schrödinger Equation with Distributed Coefficients

In this paper, we first present a complex multirational exp‐function ansatz for constructing explicit solitary wave solutions, N‐wave solutions, and rouge wave solutions of nonlinear partial differential equations (PDEs) with complex coefficients. To illustrate the effectiveness of the complex multirational exp‐function ansatz, we then consider a generalized nonlinear Schrödinger (gNLS) equation with distributed coefficients. As a result, some explicit rational exp‐function solutions are obtained, including solitary wave solutions, N‐wave solutions, and rouge wave solutions. Finally, we simulate some spatial structures and dynamical evolutions of the modules of the obtained solutions for more insights into these complex rational waves. It is shown that the complex multirational exp‐function ansatz can be used for explicit solitary wave solutions, N‐wave solutions, and rouge wave solutions of some other nonlinear PDEs with complex coefficients.

Comparison of Analytical and Numerical Simulations of Long Nonlinear Internal Solitary Waves in Shallow Water

ABSTRACT Mohapatra, S.C.; Fonseca, R.B., and Guedes Soares, C., 2018. Comparison of analytical and numerical simulations of long nonlinear internal solitary waves in shallow water. This study compares the results of long nonlinear internal waves between analytical and numerical wave models in shallow water. To present the analytical solutions, the (G′/G) expansion method is applied to obtain the exact and explicit nonlinear solitary wave solutions of the generalized Gardner equation of any order in the presence of dispersive coefficient. The internal waves are demonstrated by analysing the solutions of the classical Gardner equation, modified Korteweg-de Vries equation, and Korteweg-de Vries equations from the generalized solution. The shape and characteristics of the three-dimensional solitary waves are studied by analysing different numerical results of the nonlinear analytical solutions. The Gardner equation basically behaves with the same properties as the Boussinesq equations of travelling wave solut...

Piecewise linear approximation to Fisher's equation

A simple method is presented which allows the replacement of a nonlinear differential equation with a piecewise linear differential equation. The method is based on the idea that a curve of the nonlinear terms of the dependent variable in a differential equation can be replaced by an approximate curve consisting of a set of line segments tangent to the original curve. We apply this method to the ubiquitous Fisher's equation and demonstrate that accurate solutions are obtained with a relatively small number of line segments. References R. A. Fisher. The wave of advance of advantageous genes. Ann. of Eugenics. , 7 , 355--369, 1937. V. M. Kenkre and M. N. Kuperman. Applicability of the Fisher equation to bacterial population dynamics. Phys. Rev. E. , 67 , 2003, 051921. doi:10.1103/PhysRevE.67.051921 D. A. Kessler, Z. Ner and L. M. Sander. Front propagation: precursors, cutoffs, and structural stability. Phys. Rev. E. , 58 , 1998, 107--114. P. K. Maini, D. L. S. McElwain and D. I. Leavesley. Traveling wave model to interpret a wound-healing cell migration assay for human peritoneal mesothelial cells. Tissue Eng. , 10 , 2004, 475--482. M. J. Ablowitz and A. Zeppetella. Explicit solutions of Fisher's equation for a special wave speed. Bull. Math. Biol. , 41 , 1979, 835--840. X. Y. Wang. Exact and explicit solitary wave solutions for the generalised Fisher equation. Phys. Lett. A , 131 , 1988, 277--279. J. Rinzel and J. B. Keller. Traveling wave solutions of a nerve conduction equation. Biophys. J. , 13 , 1973, 1313--1337. Z. Jovanoski and S. Soo. Piecewise linear approximation of nonlinear ordinary differential equations. ANZIAM J. (E) , 51 , 2010, C570--C585. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/2622 T. Hagstrom and H. B. Keller. The numerical calculation of traveling wave solutions of nonlinear parabolic equations. SIAM J. Sci. Stat. Comput. , 7 , 1986, 978--988.

Application of Bifurcation Method to the Generalized Zakharov Equations

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic blow‐up wave solutions and solitary wave solutions.

Bifurcation Behaviour of the Travelling Wave Solutions of the Perturbed Nonlinear Schrödinger Equation with Kerr Law Nonlinearity

In this paper, we study the bifurcations and dynamic behaviour of the travelling wave solutions of the perturbed nonlinear Schrödinger equation (NLSE) with Kerr law nonlinearity by using the theory of bifurcations of dynamic systems. Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained

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.

Solitary wave solutions of the high-order KdV models for bi-directional water waves

An approach, which allows us to construct specific closed-form solitary wave solutions for the KdV-like water-wave models obtained through the Boussinesq perturbation expansion for the two-dimensional water wave problem in the limit of long wavelength/small amplitude waves, is developed. The models are relevant to the case of the bi-directional waves with the amplitude of the left-moving wave of O( ϵ) ( ϵ is the amplitude parameter) as compared with that of the right-moving wave. We show that, in such a case, the Boussinesq system can be decomposed into a system of coupled equations for the right- and left-moving waves in which, to any order of the expansion, one of the equations is dependent only on the (main) right-wave elevation and takes the form of the high-order KdV equation with arbitrary coefficients whereas the second equation includes both elevations. Then the explicit solitary wave solutions constructed via our approach may be treated as the exact solutions of the infinite-order perturbed KdV equations for the right-moving wave with the properly specified high-order coefficients. Such solutions include, in a sense, contributions of all orders of the asymptotic expansion and therefore may be considered to a certain degree as modelling the solutions of the original water wave problem under proper initial conditions. Those solitary waves, although stemming from the KdV solitary waves, possess features found neither in the KdV solitons nor in the solutions of the first order perturbed KdV equations.

Explicit peakon and solitary wave solutions for the modified Fornberg–Whitham equation

In this paper, the modified Fornberg–Whitham equation is studied by using the bifurcation theory and the method of phase portraits analysis. In some parametric conditions, some peakons and solitary waves are found and their exact parametric representations in explicit form are obtained.

Explicit periodic solitary wave solutions for the (2 + 1)-dimensional Boussinesq equation

In this paper, (2 + 1)-dimensional Boussinesq equation is investigated. By using homoclinic test method with the aid of Maple, new explicit periodic solitary wave solutions are obtained. Moreover, mechanical feature of wave is exhibited.

Solitary waves in coupled nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities

Using Lie group theory we construct explicit solitary wave solutions of coupled nonlinear Schrödinger systems with spatially inhomogeneous nonlinearities. We present the general theory, use it to construct different families of explicit solutions and study their linear and dynamical stability.

Generalized Kaup–Kupershmidt solitons and other solitary wave solutions of the higher order KdV equations

New types of exact explicit solitary wave solutions of the higher order KdV equations are identified by applying a direct method designed specifically for constructing solitary wave solutions of evolution equations. The first type is the ‘generalized Kaup–Kupershmidt’ (GKK) solitary waves, which unify the structures of the sech2 KdV-like soliton and the Kaup–Kupershmidt soliton and also provide solutions of some other equations. One of those equations is found to possess the multi-soliton solutions which makes it a good candidate for being integrable in terms of the GKK solitons. Another type of solutions of the higher order KdV equations identified by applying the method represents the steady-state localized structures. The variety of equations possessing such solutions includes the integrable (in terms of the sech2 solitons) Sawada–Kotera equation which thus appears to provide the localized solutions of two types.

New abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov system

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the soliton solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.

Travelling solitary wave solutions for the generalized Burgers–Huxley equation with nonlinear terms of any order

In this paper, the travelling wave solutions for the generalized Burgers–Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved.