Families Of Solitary Wave Solutions Research Articles

Exploring Families of Solitary Wave Solutions for the Fractional Coupled Higgs System Using Modified Extended Direct Algebraic Method

In this paper, we suggest the modified Extended Direct Algebraic Method (mEDAM) to examine the existence and dynamics of solitary wave solutions in the context of the fractional coupled Higgs system, with Caputo’s fractional derivatives. The method begins with the formulation of nonlinear differential equations using a fractional complex transformation, followed by the derivation of solitary wave solutions. Two-dimensional, Three-dimensional and contour graphs are used to investigate the behavior of traveling wave solutions. The research reveals many families of solitary wave solutions as well as their deep interrelationships and dynamics. These discoveries add to a better understanding of the dynamics of the fractionally coupled Higgs system and have potential applications in areas that use nonlinear Fractional Partial Differential Equations (FPDEs).

New Family of Solitary Wave Solutions to New Generalized Bogoyavlensky–Konopelchenko Equation in Fluid Mechanics

New Family of Solitary Wave Solutions to New Generalized Bogoyavlensky–Konopelchenko Equation in Fluid Mechanics

Novel soliton structures of truncated M-fractional (4+1)-dim Fokas wave model

Abstract In this research article, a nonlinear time–space fractional order (4+1)-dim Fokas wave equation that is crucial for examining the corporal marvels of waves on and inside the surface of water is examined. For this purpose, a well-known analytical method is utilized, namely, the Sardar sub-equation (SSE) method along with a truncated M-fractional derivative. As a result, many new families of solitary wave solutions, such as kink-type solitons, singular and periodic solitons, dark and bright solitons, are established. By using the SSE method, the outcomes are portrayed in 3-dim, 2-dim, and contour plots for distinct parametric values. The attained hyperbolic and trigonometric function-type results demonstrate the capability of recognizing the exact solutions of the other nonlinear evolution equations through the executed technique.

New solitary wave solutions of the Korteweg-de Vries (KdV) equation by new version of the trial equation method

New solitary wave solutions for the Korteweg-de Vries (KdV) equation by a new version of the trial equation method are attained. Proper transformation reduces the Korteweg-de Vries (KdV) equation to a quadratic ordinary differential equation that is fully integrated using the new version trial equation approach. The family of solitary wave solutions of the reduced equation ensures a combined expression for the Korteweg-de Vries (KdV) equation, which contains exact solutions derived in recent years using different integration methods. The analytic solution of the reduced equation permits to find exact solutions for the Korteweg-de Vries (KdV) equation, providing a variety of new solitary wave solutions that have not been reported before.

Asymptotic stability of solitary waves for the <inline-formula><tex-math id="M1">$ 1d $</tex-math></inline-formula> NLS with an attractive delta potential

We consider the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases. We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result [35] beyond the regime of small solitary waves and extend the results of [19,29] from orbital to asymptotic stability for a suitable family of solitary waves.

Closed-Form Solutions in a Magneto-Electro-Elastic Circular Rod via Generalized Exp-Function Method

In this study, the dispersal caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod is taken into consideration using the nonlinear longitudinal wave equation (LWE), a mathematical physics problem. Using the generalized exp-function method, we investigate the families of solitary wave solutions of one-dimensional nonlinear LWE. Using the computer program Wolfram Mathematica 10, these new exact and solitary wave solutions of the LWE are derived as trigonometric function, periodic solitary wave, rational function, hyperbolic function, bright and dark solitons solutions, sinh, cosh, and sech2 function solutions of the LWE. These solutions represent the electrostatic potential and pressure for LWE as well as the graphical representation of electrostatic potential and pressure.

Dispersive analytical wave solutions and abundant closed‐form wave solutions of some nonlinear dynamical models arising in fluid mechanics with Stability analysis

In this manuscript, we have employed improved form of extended simple equation and modified F‐expansion to construct the stability analysis solitary wave solution of simplified modified forms of Camassa‐Holm equation, the Newell‐Whitehead‐Segel, and Zeldovich Equations. The novel families of solitary wave solutions are in the form of expontial, trigonometric, and hyperbolic functions, respectively, with kink and anti ‐kink solitons; bright and dark solitons are obtained. We also give analysis for stability of our governing model. Some results are displayed graphically in the form of 3D and 2D to explain the physical behavior of concern models. The investigated results show the power and fruitfulness of our presented schemes and solutions play important rule in different field of applied science.

The existence and decay of solitary waves for the Fornberg–Whitham equation

In this paper, we consider the Fornberg-Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in our proof. Besides, we also prove that the family of solitary solutions is orbital stable and decay exponentially when speed wave c is bigger than 1.

A New Type of Solitary Wave Solution of the mKdV Equation Under Singular Perturbations

In this work, we examine the solitary wave solutions of the mKdV equation with small singular perturbations. Our analysis is a combination of geometric singular perturbation theory and Melnikov’s method. Our result shows that two families of solitary wave solutions of mKdV equation, having arbitrary positive wave speeds and infinite boundary limits, persist for selected wave speeds after small singular perturbations. More importantly, a new type of solitary wave solution possessing both valley and peak, named as breather in physics, which corresponds to a big homoclinic loop of the associated dynamical system is observed. It reveals an exotic phenomenon and exhibits rich dynamics of the perturbed nonlinear wave equation. Numerical simulations are performed to further detect the wave speeds of the persistent solitary waves and the nontrivial one with both valley and peak.

Instability of solitary wave solutions for the nonlinear Schrödinger equation of derivative type in degenerate case

We study the stability theory of solitary wave solutions for a type of derivative nonlinear Schrödinger equation i∂tu+∂x2u+i|u|2∂xu+b|u|4u=0,b>0.The equation has a two-parameter family of solitary wave solutions of the form uω,c(x,t)=exp{iωt+ic2(x−ct)−i4∫−∞x−ct|φω,c(η)|2dη}φω,c(x−ct).Here φω,c is some suitable function and −2ω<c≤2ω. The stability in the frequency region of −2ω<c<2κω (for some κ∈(0,1)), and the instability in the frequency region of 2κω<c<2ω were proved by Ohta (2014). Recently, in the endpoint case c=2ω, the instability of uω,c was proved by Ning et al. (2017). Then the stability and instability region has been established except the degenerate case c=2κω. In this paper, we address the problem and prove its instability in the degenerate case.

Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev–Petviashvili modified equal width dynamical equation

In this research work, we constructed the solitary wave solutions of generalized Kadomtsev–Petviashvili modified equal width (KP-MEW) equation with the help of new technique which is modification form of extended auxiliary equation mapping method. The generalized KP-MEW equation is the nonlinear PDEs which described the propagation of long-wave with dissipation and dispersion in nonlinear media. As a result, families of solitary wave solutions are obtained in different form of solitons, bright–dark solitons and traveling wave solutions. The physical structure of these new solutions is shown graphically in two and three dimensions with the aid of computer software Mathematica. These obtained new solutions show the power and effectiveness of this new method. We can also solve other nonlinear system of PDEs which are involved in mathematical physics and many other branches of physical sciences with the help of this new method.

Applications of nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod and new solitary wave solutions

In this work, we consider the nonlinear longitudinal wave equation (LWE) which involves mathematical physics with dispersal produced by the phenomena of transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod. We use the extended form of two methods, auxiliary equation mapping and direct algebraic method to investigated the families of solitary wave solutions of one-dimensional nonlinear LWE. These new exact and solitary wave solutions are derived in the form of trigonometric function, periodic solitary wave, rational function, and elliptic function, hyperbolic function, bright and dark solitons solutions of the LWE, which represent the electrostatic potential and pressure for LWE and also the graphical representation of electrostatic potential and pressure are shown with the aid of Mathematica program.

Propagation of nonlinear complex waves for the coupled nonlinear Schrödinger Equations in two core optical fibers

In this paper we discussed analytically integrable coupled nonlinear Schrödinger Equation with Kerr law nonlinearity with the aid of newly developed technique named as extended modified auxiliary equation mapping method. As a result of which we have found a variety of new families of solitary wave solutions including bright, dark, half bright, half dark, combined, periodic, doubly periodic with the help of three parameters which is the key importance of this method. For physical description of our newly obtained solutions we have expressed them graphically using Mathematica 10.4 to explain more efficiently the behavior of different shapes of solutions.

Propagation of long-wave with dissipation and dispersion in nonlinear media via generalized Kadomtsive–Petviashvili modified equal width-Burgers equation

The generalized Kadomtsive–Petviashvili modified equal width-Burgers (KP-MEW-B) equation described the propagation of long-wave with dissipation and dispersion in nonlinear media. We investigated the solitary wave solutions of generalized KP-MEW-B equation by applying modification form of extended auxiliary equation mapping method. As a results, families of solitary wave solutions are obtained in different form of solitons: the single bright–dark solitons, the double bright–dark solitons and traveling wave solutions. The physical structure of these new solutions are shown in two and three dimensional graphically with the aid of computer software Mathematica. These obtained new solutions show the power and effectiveness of this new method. We can also solve other unstable nonlinear system of PDEs which are involved in mathematical physics and many other branches of physical sciences with the help of this new method .

Some new families of solitary wave solutions of the generalized Schamel equation and their applications in plasma physics

In this article we studied analytically the propagation of nonlinear ion acoustic solitary waves modeled by the generalized Schamel (GS) equation arising in plasma physics using auxiliary equation mapping method. As a result, we found a series of more general and new families of solutions, which are more powerful in the development of soliton dynamics, quantum plasma, adiabatic parameter dynamics, biomedical problems, fluid dynamics, industrial studies and many other fields. The calculations prove that this method is more reliable, straightforward, and effective to study analytically other nonlinear complicated physical problems modeled by complex nonlinear partial differential equations arising in mathematical physics, hydrodynamics, fluid mechanics, mathematical biology, plasma physics, engineering disciplines, chemistry and many other natural sciences. We also have expressed our solutions graphically with the help of Mathematica 10.4 to understand physically the behavior of different shapes of ion acoustic solitary waves including kink-type, anti-kink-type, half-bright and dark soliton.

Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations

AbstractIn this research work, for the first time we introduced and described the new method, which is modified extended auxiliary equation mapping method. We investigated the new exact traveling and families of solitary wave solutions of two well-known nonlinear evaluation equations, which are generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified forms of Camassa-Holm equations. We used a new technique and we successfully obtained the new families of solitary wave solutions. As a result, these new solutions are obtained in the form of elliptic functions, trigonometric functions, kink and antikink solitons, bright and dark solitons, periodic solitary wave and traveling wave solutions. These new solutions show the power and fruitfulness of this new method. We can solve other nonlinear partial differential equations with the use of this method.

Vanishing characteristic speeds and critical dispersive points in nonlinear interfacial wave problems

Criticality plays a central role in the study of reductions and stability of hydrodynamical systems. At critical points, it is often the case that nonlinear reductions with dispersion arise to govern solution behavior. By considering when such models become bidirectional and lose their initial dispersive properties, it will be shown that higher order dispersive models may be supported in hydrodynamical systems. Precisely, this equation is a two-way Boussinesq equation with sixth order dispersion. The case of two layered shallow water is considered to illustrate this, and it is reasoned why such an environment is natural for such a system to emerge. Further, it is demonstrated that the regions in the parameter space for nontrivial flow, which admit this reduction, are vast and in fact form a continuum. The reduced model is then numerically simulated to illustrate how the two-way and higher dispersive properties suggest more exotic families of solitary wave solutions can emerge in stratified flows.

Instability of solitary wave solutions for derivative nonlinear Schrödinger equation in endpoint case

We study the stability theory of solitary wave solutions for a type of the derivative nonlinear Schrödinger equationi∂tu+∂x2u+i|u|2∂xu+b|u|4u=0. The equation has a two-parameter family of solitary wave solutions of the formeiω0t+iω12(x−ω1t)−i4∫−∞x−ω1t|φω(η)|2dηφω(x−ω1t). The stability theory in the frequency region of |ω1|<2ω0 was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case ω1=2ω0, in which the elliptic equation of φω is “zero mass”.

EXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT EQUATION

In this paper we investigate the exact traveling wave solutions of the fifth-order Kaup-Kuperschmidt equation. The bifurcation and exact solutions of a general first-order nonlinear equation are investigated firstly. With the help of Maple and by using the bifurcation and exact solutions of two derived subequations, we obtain two families of solitary wave solutions and two families of periodic wave solutions of the KK equation. The relationship of the two subequations and the two known rst integrals are analyzed.

Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrödinger Equation

We consider a derivative nonlinear Schr\"odinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, their velocity. We illustrate these results with numerical simulations.