Singular Solitary Wave Research Articles

Multiple tsunami waves of the fractional geophysical KdV equation

Exact analytical form and approximate form of different kinds of tsunami wave solutions of the fractional geophysical Korteweg-de-Vries (fgKdV) equation are derived. The fgKdV equation is constructed by replacing the Caputo fractional derivative from the geophysical Korteweg-de-Vries equation. Some important definitions and results of the Caputo fractional derivative are discussed. The Darboux transformation and Residual power series method are used to derive some exact solutions and approximate analytical solutions to the fgKdV equation. Using the Darboux transformation the single solitary tsunami wave, singular periodic tsunami wave, two-soliton tsunami wave, singular solitary tsunami wave, solitary periodic tsunami wave, rational tsunami wave, rational solitary tsunami wave, and rational periodic tsunami wave were obtained. By performing the Residual power series method the approximate analytical solution is obtained. The graphical presentations of several exact solutions are obtained by Darboux transformation and Residual power series method. It is seen that the fractional parameter has a significant effect on the structure of the nonlinear tsunami waves of the fgKdV equation. The results of this study are applicable to understand the dynamics of multiple tsunami wave described by the fgKdV equation in the ocean.

Diverse exact solutions to Davey–Stewartson model using modified extended mapping method

In this study, we obtain solitary wave solutions and other exact wave solutions for Davey–Stewartson equation (DSE), which explains how waves move through water with a finite depth while being affected by gravity and surface tension. The study is conducted with the aid of the modified extended mapping method (MEMM). A variety of distinct traveling wave solutions are furnished. The obtained solutions comprise dark, bright, and singular solitary wave solutions. Additionally, Jacobi elliptic function solutions, exponential wave solutions, singular periodic wave solutions, rational wave solutions, and periodic wave solutions are also offered. To help readers physically grasp the acquired solutions, graphical representations of some of the extracted solutions are provided.

The new kink type and non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

In this paper, the new solitary wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation are obtained by Lie group symmetry method and the extended homoclinic test approach. Firstly, the equation can be reduced to (1+1)-dimensional partial differential equation by Lie group symmetry, and corresponding bilinear forms of the equation are given by symmetry functions. Secondly, the extended homoclinic test approach is employed to obtain the new kink type and singular solitary wave solutions. In addition, some new traveling and non-traveling wave solutions with arbitrary functions and oscillating tail are investigated through the special transformations for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation.

New exact solutions to the generalized shallow water wave equation

In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this nonlinear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation flows. To obtain the new exact solutions of the considered model, we have applied a novel analytical technique namely [Formula: see text]-expansion method. Using the aforementioned method and computational software, we have obtained different kinds of periodic and singular solitary wave solutions of the generalized shallow water wave equation. The obtained solutions are exponential function and trigonometric function solutions. Using 2D and 3D plots of the wave solutions, the dynamic behaviors of the developed solutions are displayed. The retrieved solutions validated the effectiveness and robustness of the proposed technique.

Diversity of soliton solutions to the (3 + 1)-Dimensional Wazwaz-Benjamin-Bona-Mahony equations arising in mathematical physics

The current study scrutinizes exact singular solitary, kink, anti-kink, bell-shaped, and periodic traveling wave solutions for the newly implemented 3-dimensional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations family using a modified Exp-function approach. In spite of the presence of several trigonometric, hyperbolic, and rational functions, some new exact solitary, hyperbolic, and periodic traveling wave solutions to the considered equations are obtained by putting the modified Exp-function method into practice using the computer program Maple. The arrangement of the functions tanh, sech2, tan, and sec2 expresses the specific solutions produced by the defined procedure. The information was obtained to evaluate how well the suggested method could compute the exact solutions of the WBBM equations that could be applied to nonlinear water model applications in the ocean and coastal engineering. To show the physical composition and properties of the discovered solitons, contour, three-, and two-dimensional graphs are plotted using the computer program Mathematica. The acquired results imply that the proposed method can be applied to find various, enhanced, useful, and compatible solutions for other important nonlinear partial differential equations. All of the solutions given have been qualified by swapping the respective equations with those generated via the computer program Maple. Furthermore, we compared our obtained results with the solutions already put forward in the literature.

Some new kink type solutions for the new (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

Exact solutions of higher-dimensional nonlinear equations takes a major place in the study of nonlinear phenomena observed in nature. In this article, some new kink type solutions are investigated for the new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli(BLMP) equation. Firstly, a variety of solutions are obtained by Hirota’s bilinear form, which include kink type wave solution, periodic solitary wave solutions and singular solitary wave solutions using extended homoclinic test approach. Secondly, solutions with three wave form are obtained by generalized three wave method. The extended homoclinic test approach is also used to construct solutions with a tail which explain some physical phenomenon. Moreover, some figures of the solutions are shown behind.

Novel solitary waves for fractional (2+1)-dimensional Heisenberg ferromagnetic model via new extendedgeneralized Kudryashov method

This article introduces the fractional (2+1)-dimensional evolution model portraying the transmission of nonlinear spin dynamics in Heisenberg ferromagnetic spin chains with bilinear and anisotropic interactions in the semiclassical limit. Fractional derivatives stand out in modeling problems involving the concepts of non-locality and memory effect, that are not well explained by the integer-order calculus. New extended generalized Kudryashov method is employed to construct traveling wave solutions of governing equation. These solutions such as dark, singular solitons, solitary waves and singular solitary waves arise with constraint conditions and exhibit distinct physical significance. Since magnetic soliton has been classified as one of the fascinating groups of nonlinear excitations expressing spin dynamics in semiclassical continuum Heisenberg systems. 3D and 2D graphs for suitable parametric values are provided to interpret the dynamics of soliton profiles. Also, a comparative analysis is shown among Beta and Atangana-Baleanu fractional derivatives. Our results assert that this scheme is reliable and efficient gadget for extracting the soliton solutions of different nonlinear evolution equations appearing in mathematical sciences.

Solitary waves with intensity-dependent dispersion: variational characterization

A continuous family of singular solitary waves exists in a prototypical system with intensity-dependent dispersion. The family has a cusped soliton as the limiting lowest energy state and is formed by the solitary waves with bell-shaped heads of different lengths. We show that this family can be obtained variationally by minimization of mass at fixed energy and fixed length of the bell-shaped head. We develop a weak formulation for the singular solitary waves and prove that they are stable under perturbations which do not change the length of the bell-shaped head. Numerical simulations confirm the stability of the singular solitary waves.

Harmonizing wave solutions to the Fokas-Lenells model through the generalized Kudryashov method

In this article, the closed form general and standard solutions accessible in the literature of nonlinear evolution equation (NLEE), namely, the Fokas-Lenells (FL) equation is established by using the generalized Kudryashov approach. The implemented method extracts lots of new and compatible wave solutions involving unknown parameters and expressed in terms of exponential, trigonometric and hyperbolic functions. We explain the physical meaning of geometrical structures for some derived solutions by assigning definite values of the unspecified parameters. As an aftermath, diverse wave shapes including kink wave shape, flat kink wave shape, bell shape, anti-bell shape, singular bell shape along with bright, dark, singular solitary wave envelopes are properly developed. These nonlinear wave envelopes are frequently applicable in optical fibers. It has been established that the assigned techniques are general, efficient and straightforward and also it can be exerted to accomplish closed form solutions of diverse NLEEs originated in mathematical physics and engineering fields.

Exact solutions of the combined Hirota-LPD equation with variable coefficients

In this paper, we construct exact families of traveling wave (periodic wave, singular wave, singular periodic wave, singular-solitary wave and shock wave) solutions of a well-known equation of nonlinear PDE, the variable coefficients combined HirotaLakshmanan-Porsezian-Daniel (Hirota-LPD) equation with the fourth nonlinearity, which describes an important development, and application of soliton dispersion management experiment in nonlinear optics is considered, and as an achievement, a series of exact traveling wave solutions for the aforementioned equation is formally extracted. This nonlinear equation is solved by using the extended trial equation method (ETEM) and the improved tan(ϕ/2)-expansion method (ITEM). Meanwhile, the mechanical features of some families are explained through offering the physical descriptions. Analytical treatment to find the nonautonomous rogue waves are investigated for the combined Hirota-LPD equation.

New non-traveling wave solutions for (3+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation

<abstract><p>In this paper, we investigate non-traveling wave solutions of the (3+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa (VC-DJKM) equation, which describes the real physical phenomena owing to the inhomogeneities of media. By combining the extended homoclinic test approach with variable separation method, we obtain abundant new exact non-traveling wave solutions of the (3+1)-dimensional VC-DJKM equation. These results with a parabolic tail or linear tail reveal the complex structure of the solutions for (3+1)-dimensional VC-DJKM equation. Moreover, the tail in these solutions maybe give a prediction of physical phenomenon. When arbitrary functions contained in these non-traveling wave solutions are taken as some special functions, we can get the kink-type solitons, singular solitary wave solutions, and periodic solitary wave solutions, and so on. As the special cases of our work, the corresponding results of (3+1)-dimensional DJKM equation, (2+1)-dimensional DJKM equation, (2+1)-dimensional VC-DJKM equation are also given.</p></abstract>

ABUNDANT NEW NON-TRAVELING WAVE SOLUTIONS FOR THE (3+1)-DIMENSIONAL BOITI-LEON-MANNA-PEMPINELLI EQUATION

<abstract> Seeking exact solutions of higher-dimensional nonlinear partial differential equations has recently received tremendous attention in mathematics and physics. In this paper, we investigate exact solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation which describes nonlinear wave propagation in incompressible fluid. Firstly, by means of extended homoclinic test approach, we get eight kinds of non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Then, combining the improved tanh function method and new ansatz solutions, we obtain abundant new exact non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. These results include not only many results obtained in other literatures, but also some new exact non-traveling wave solutions. Moreover, the exact kink wave solutions, periodic solitary wave solutions and singular solitary wave solutions are given when arbitrary functions contained in these solutions are taken as some special functions. </abstract>

Construction of traveling waves patterns of [formula omitted]-dimensional modified Zakharov-Kuznetsov equation in plasma physics

In this research, we examine the modified model of (1+n)-dimensional Zakharov-Kuznetsov (ZK) equation, which will be used to analyze the nature of weakly nonlinear traveling waves in the existence of a constant magnetic area in a plasma comprising in cold ions and hot isothermal electrons. The modified Zakharov-Kuznetsov (mZK) equation will have solutions describing the traveling solitary waves, using the extended (G′G2)-expansion method and extended direct algebraic method gives way to the mZK equation regulating the transmission of ion dynamics for nonlinear traveling waves in a plasma. The sufficient conditions for the stability and existence of the traveling wave solutions are reported. Semi-dark, rational, and singular solitary wave solutions are computed. Graphical interpretations of certain practical solutions for specific values of parameters have also been available. The research findings reported throughout this evaluation are fresh and from which this model is employed to analyze waves in numerous plasmas, could be valuable and important. Subsequently, there are concluding remarks mentioned.

On traveling wave solutions: The Wu–Zhang system describing dispersive long waves

In this paper, we construct exact families of traveling wave (periodic wave, solitary wave, shock wave, singular-wave, singular-periodic wave, and singular-solitary wave) solutions of a well-known system of nonlinear PDEs, the Wu–Zhang system, which describes (1[Formula: see text]+[Formula: see text]1)-dimensional dispersive long waves. This system is solved by using the generalized [Formula: see text] expansion method, where G satisfies the Jacobi elliptic equation of fourth order. Meanwhile, the mechanical features of some families are explained through three-dimensional figures.

Optical and singular solitary waves to the PNLSE with third order dispersion in Kerr media via two integration approaches

In this paper, the sine–Gordon expansion method (SGEM) and the generalized tanh (GTM) methods are employed to find the optical solitary wave solutions to the perturbed nonlinear Schrödinger equation (PNLSE) with Kerr law nonlinearity describing the propagation of optical solitary waves in nonlinear optical fibers. The dark, bright, dark–bright and dark-singular optical solitary waves are retrieved. Additionally, singular solitary waves are also derived as by-products of the integration scheme. Physical interpretations of the obtained results are demonstrated with interesting figures. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the PNLSE.

Novel optical solitary waves and modulation instability analysis for the coupled nonlinear Schrödinger equation in monomode step-index optical fibers

This paper addresses the coupled nonlinear Schrödinger equation (CNLSE) in monomode step-index in optical fibers which describes the nonlinear modulations of two monochromatic waves, whose group velocities are almost equal. A class of dark, bright, dark-bright and dark-singular optical solitary wave solutions of the model are constructed using the complex envelope function ansatz. Singular solitary waves are also retrieved as bye products of the in integration scheme. This naturally lead to some constraint conditions placed on the solitary wave parameters which must hold for the solitary waves to exist. The modulation instability (MI) analysis of the model is studied based on the standard linear-stability analysis. Numerical simulation and physical interpretations of the obtained results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the CNLSE.

Solitons and other solutions to Wu–Zhang system

This paper addresses Wu–Zhang system to study dispersive long waves. The extended trial equation method extracts solitary waves, shock waves and singular solitary waves solutions. Subsequently, Lie group formalism is also applied to derive symmetries of the Wu–Zhang system and the derived ordinary differential equations are further analyzed to retrieve exact solutions are obtained. Finally, implementation of mapping method secures additional exact solutions.

Group analysis, nonlinear self-adjointness, conservation laws, and soliton solutions for the mKdV systems

We study the symmetry groups, conservation laws, solitons, and singular solitary waves of some versions of systems of the modified KdV equations.

Jacobi elliptic solutions, soliton solutions and other solutions to four higher-order nonlinear Schrodinger equations using two mathematical methods

Two mathematical methods, via the (G′/G)-expansion method and extended auxiliary equation method are applied in this article to find many exact solutions with parameters of four higher-order nonlinear Schrodinger equations whose balance numbers are not positive integers, namely, the nonlinear Schrodinger equation with dual power-law nonlinearity, the higher-order dispersive nonlinear Schrodinger equation with both fourth-order dispersion effects and a quintic nonlinearity, the resonant nonlinear Schrodinger equation and the generalized higher-order nonlinear Schrodinger equation. When the parameters take up special values, the solitary and singular solitary wave solutions of these nonlinear Schrodinger equations are given. The used methods in this article present a wider applicability for handling nonlinear wave equations. Comparing our new results and the well-known results are obtained. Comparing the results resulting from the two methods with each others are also given.

On soliton solutions of the Wu-Zhang system

AbstractIn this paper, the extended tanh and hirota methods are used to construct soliton solutions for the WuZhang (WZ) system. Singular solitary wave, periodic and multi soliton solutions of the WZ system are obtained.