New Families Of Exact Solutions Research Articles

Optical Solitons and traveling wave solutions to Kudryashov’s equation

This paper employs a generalized transformation to investigate Kudryashov’s equation with four nonlinear terms in order to construct traveling wave solutions and optical solitons. The proposed unified ansätze framework is manipulated to derive analytical solutions, which are expressed as a combination of general solutions to simpler equations or systems of equations that are integrable or have unique known solutions. Using this technique, one can produce families of new exact solutions as well as solitary solutions. In addition to other model solutions, the scheme reveals the elliptic functions of Jacobi, Weierstrass, and Riccati. These are categorized as singular, bright, and dark optical solitons.

Semi-analytical solutions of shallow water waves with idealised bottom topographies

Analysing two-dimensional shallow water equations with idealised bottom topographies have many applications in the atmospheric and oceanic sciences; however, restrictive flow pattern assumptions have been made to achieve explicit solutions. This work employs the Adomian decomposition method (ADM) to develop semi-analytical formulations of these problems that preserve the direct correlation of the physical parameters while capturing the nonlinear phenomenon. Furthermore, we exploit these techniques as reverse engineering mechanisms to develop key connections between some prevalent ansatz formulations in the open literature as well as derive new families of exact solutions describing geostrophic inertial oscillations and anticyclonic vortices with finite escape times. Our semi-analytical evaluations show the promise of this approach in terms of providing robust approximations against several oceanic variations and bottom topographies while also preserving the direct correlation between the physical parameters such as the Froude number, the bottom topography, the Coriolis parameter, as well as the flow and free surface behaviours. Our numerical validations provide additional confirmations of this approach while also illustrating that ADM can also be used to provide insight and deduce novel solutions that have not been explored, which can be used to characterise various types of geophysical flows.

Several families of new exact solutions for second order partial differential equations with variable coefficients

Several families of new exact solutions for a general second order linear partial differential equation with variable coefficients are derived in this paper. All the possible polynomial and polynomial-like solutions of this equation are derived. It is shown that there exist exactly two sets of such families of exact solutions. These solutions are extended to construct different families of exact solutions in terms of hypergeometric functions, which include polynomial solutions as particular cases. A total of eight families of exact solutions are derived using a novel method of balancing powers of the variables simultaneously. Several well known linear partial differential equations in applied mathematics and mechanics are special cases of the general equation considered in this paper and all the polynomial and polynomial-like solutions of these partial differential equations are also explicitly derived as special cases.

On traveling wave solutions of an autocatalytic reaction–diffusion Selkov–Schnakenberg system

This paper investigates the analytical solutions of the autocatalytic reaction–diffusion Selkov–Schnakenberg system of coupled nonlinear PDEs mathematical perspective. This is a simple chemical reaction system that admits periodic solutions. The new families of exact solutions are derived which represent the autocatalyst and reactant of chemical concentrations. These exact solutions are obtained by using the technique namely as ϕ6-model expansion. Furthermore, the existence of these solutions is also discussed under different constraint conditions and variables of concentrations that are represented in hyperbolic, trigonometric, and rational forms. These results are new and effective in the physical phenomena of autocatalytic chemical reactions. For a better understanding of the physical interpretation of the solutions, the 3D and 2D graphs of some reported solutions are dispatched below for the different choices of parameters. Hence the physical description of our results may fruitful tool for investigating the further results for nonlinear wave problems in applied science.

New classes of periodic and non-periodic exact solutions for Newtonian and non-Newtonian fluid flows

Two different families of planar exact solutions for second grade fluid flows are derived in this paper. Exact solutions for Newtonian fluid flows are derived as particular case when normal stress moduli vanish. The novelty of the solutions is that they are obtained by assuming the stream functions as a finite sum of functions with different arguments. An important property of the derived solutions is that solutions in a given family are all superposable flows. The first family of solutions can be considered as a generalization of rectangular array of Taylor vortices as these classical solutions are special cases of the derived exact solutions. A subfamily of spatially periodic exact solutions for both Newtonian and non-Newtonian fluid flows are also derived. The complex structures of the two dimensional vortices are illustrated in several cases. • Two new families of exact solutions for second grade fluid flows are derived. • These solutions are classified as generalized Taylor vortices. • The solutions in a given family are superposable flows. • Subfamily of spatially periodic solutions are constructed. • Exact solutions for Newtonian flows are also derived.

Demonstration of unique problems from Soliton solutions to nonlinear Selkov–Schnakenberg system

This article investigates the existence theory, exact solutions, and the unique solutions of physical problems. In this study the well-known Selkov–Schnakenberg system of coupled nonlinear unidirectional PDEs is analyzed. This is a simple chemical reaction system that admits periodic solutions. The existence of the system is extracted by applying contraction and self-mapping conditions. The new families of exact solutions which represent the concentration and chemical reactants in different forms are formulated. The periodic, singular periodic, shock wave, singular wave, and complex solitary-shock, shock-singular, double singular, and periodic-singular solutions are successfully extracted by using the new modified extended direct algebraic (MEDA) technique. Additionally the unique problems with corresponding proposed auxiliary data are discussed. The trends of these solutions are also sketched in different plots.

New traveling wave exact solutions to the coupled Klein–Gordon system of equations

A system of coupled Klein–Gordon nonlinear partial differential equations are considered in this paper and several new families of exact solutions for this system are derived. Since this system is highly nonlinear, exact solutions are very difficult to obtain. A rational ansatz form is assumed to derive the required exact solutions in terms of the doubly periodic functions. We derive all the possible real solutions having this ansatz form and which are functions of Jacobi elliptic functions. Particular solutions corresponding to each family of solutions are presented graphically.

Soliton behavior of algae growth dynamics leading to the variation in nutrients concentration

This paper provides the effects of nutrients concentrations on the algae growth via solitary wave analysis and classical existence theory. There is a well known Diffusive Nutrient-Algae model based on the Sanyang wetland, is under consideration. The new families of exact solutions which represent nutrient concentration and algae density in different form are observed using the ϕ6-model expansion. Furthermore, the existence of these solutions are also discussed under different restrictive conditions and variables of nutrient concentration and algae density are represented in hyperbolic, trigonometric and rational forms. The graphics of these solutions are also sketched in 3D and 2D representations.

New, Spherical Solutions of Non-Relativistic, Dissipative Hydrodynamics.

We present a new family of exact solutions of dissipative fireball hydrodynamics for arbitrary bulk and shear viscosities. The main property of these solutions is a spherically symmetric, Hubble flow field. The motivation of this paper is mostly academic: we apply non-relativistic kinematics for simplicity and clarity. In this limiting case, the theory is particularly clear: the non-relativistic Navier–Stokes equations describe the dissipation in a well-understood manner. From the asymptotic analysis of our new exact solutions of dissipative fireball hydrodynamics, we can draw a surprising conclusion: this new class of exact solutions of non-relativistic dissipative hydrodynamics is asymptotically perfect.

Different families of new exact solutions for planar and nonplanar second grade fluid flows

Three different families of new exact solutions for second grade fluid flows are derived in this paper. First family of two dimensional exact solutions are obtained by assuming a novel form for the stream function and these solutions contain spatially periodic as well as non-periodic solutions. The subclass of periodic solutions extends the rectangular array of Taylor vortices to more complex structures of two dimensional vortices. Second family of planar solutions are also having vortex structures similar to Taylor vortices. Third family of exact solutions are pseudo planar flows of second kind. All these exact solutions reduce to corresponding new exact solutions for Newtonian fluid flows when the normal stress moduli are absent.

Family of new exact solutions for longitudinally expanding ideal fluids

We report on the discovery of new analytical solutions of the equations of relativistic ideal hydrodynamics. In this solution, the fluid expands in the longitudinal direction and contains a plateau structure that extends over a finite range in rapidity and can be either symmetric or asymmetric in that variable. We further calculate the corresponding pseudo-rapidity distribution of hadron yields, and find decent agreement with experimental measurements in high-energy Pb+Pb, Au+Au, p+Pb, and d+Au collisions.

Anisotropic compact stars via embedding approach in general relativity: new physical insights of stellar configurations

The main focus of this paper is to explore the possibility of providing a new family of exact solutions for suitable anisotropic spherically symmetric systems in the realm of general relativity involving the embedding spherically symmetric static metric into the five-dimensional pseudo-Euclidean space. In this regard, we ansatz a new metric potential lambda (r), and we obtained the other metric potential nu (r) by mains of embedding class one approach. The unknown constants are determined by the matching of interior space-time with the Schwarzschild exterior space-time. The physical acceptability of the generating celestial model for anisotropic compact stars is approved via acting several physical tests of the main salient features viz., energy density, radial and tangential pressures, anisotropy effect, dynamical equilibrium, energy conditions, and dynamical stability, which are well-compared with experimental statistics of four different compact stars: PSR J1416-2230, PSR J1903+327, 4U 1820-30 and Cen X-3. Conclusively, all the compact stars under observations are realistic, stable, and are free from any physical or geometrical singularities. We find that the embedding class one solution for anisotropic compact stars is viable and stable, plus, it provides circumstantial evidence in favor of super-massive pulsars.

New traveling wave rational form exact solutions for strain wave equation in micro structured solids

Strain wave equation is a fourth order non-linear partial differential equation that arises in the study of non-dissipative strain wave propagation in micro structured solids. This equation also represents the dynamics of several physical phenomena. This equation can also be consider as a generalization of Boussinesq equation with dual dispersion. In this paper, a general strain wave equation is considered and obtained several new exact solutions. A variant of F-expansion method is applied to obtain the required solutions. The available traveling wave exact solutions are primarily obtained by integrating the resulting fourth order ordinary differential equation twice. But, in this paper, we show that there exist several traveling wave solutions to strain wave equation which cannot be derived using the existing methods. Several families of new exact solutions in rational function form are derived using this novel method, without performing the initial integration.

Some new families of exact solutions to a new extension of nonlinear Schrödinger equation

Determining the exact solution to the partial differential equations has been one of the most important concerns of scientists in the various centuries. This paper applies the generalized exponential rational function method to a new extension of nonlinear Schrödinger equation. Many new analytical solutions are retrieved by choosing suitable coefficients of parameters under different family cases. Some important surfaces of results such as the imaginary part, the real part, and their modulus are also depicted with the help computational packet program. According to the results obtained in this paper, the method can be assumed to be a suitable tool in solving differential equations.

Generalized horizon functions for radiating matter

In this study, we consider a general spherically symmetric radiating matter distribution containing the cosmological constant and electric charge. We show that the boundary condition for the bounded matter distribution relating the radial pressure to the heat flow is a Riccati equation. We find a new transformation for the stellar boundary condition called the generalized horizon function since it is related to the formation of horizons during collapse. The generalized horizon function leads to a new equation which preserves the structure of the Riccati equation that governs the evolution of the boundary. Several new families of exact solutions, both implicit and explicit, to the transformed boundary condition are found. It is important to observe that the horizon function can be given explicitly in all cases. Results of previous studies arise as special cases of our general analysis.

Generalized geodesic radiating models

The Einstein field equations and junction conditions are studied for a radiating object in which particles are travelling in geodesic motion. We consider the general case including the effects of gravity, the cosmological constant and the electromagnetic field. The boundary condition is shown to be a Riccati equation in general. A transformation reduces the boundary condition to a simpler equation. Several families of new exact solutions are found, both explicitly and implicitly. The exact solutions can be written in terms of elementary functions, elliptic integrals and Gaussian hypergeometric functions. We find that the cosmological constant and charge affects the gravitational behaviour of the model. We identify earlier models as special cases in this analysis.

Steady point vortex pair in a field of Stuart-type vorticity

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$, where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

Generalized anisotropic models for conformal symmetry

We find a new family of exact solutions to the Einstein system of equations with an anisotropic fluid distribution for a spherically symmetric spacetime containing a conformal Killing vector. Simple analytic expressions describe the matter variables and the metric functions which are regular at the centre and the interior of the body. We demonstrate that the new class of exact solutions is physically reasonable and may be utilized to model a compact object. A detailed graphical analysis of the matter variables shows that the criteria for physical acceptability are satisfied. The energy conditions are satisfied, causality is not violated, and the body is stable in terms of cracking, the Harrison–Zeldovich–Novikov stability criterion, and the adiabatic index inequality. It is, therefore, possible to geometrically describe a compact object with a conformal symmetry for an astrophysical application.

More general families of exact solitary wave solutions of the nonlinear Schrödinger equation with their applications in nonlinear optics

In this article we analytically studied the complex nonlinear Schrodinger equation with Kerr law nonlinearity using the auxiliary equation mapping method, as a result, we found a series of more general and new families of exact solutions, which are more powerful in the development of soliton dynamics, quantum plasma, adiabatic parameter dynamics, biomedical problems, fluid dynamics, industrial studies, nonlinear optics and many other fields. The calculations demonstrate that this method is more reliable, straightforward and effective to analytically study 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 have also expressed our solutions graphically with the help of Mathematica 10.4 to physically understand the behavior of different shapes of solutions including kink-type, anti-kink-type, half-bright and dark solitons.

Noether symmetries and exact solutions of an Euler–Bernoulli beam model

In this paper, a Noether symmetry analysis is carried out for an Euler–Bernoulli beam equation via the standard Lagrangian of its reduced scalar second-order equation which arises from the standard Lagrangian of the fourth-order beam equation via its Noether integrals. The Noether symmetries corresponding to the reduced equation is shown to be the inherited Noether symmetries of the standard Lagrangian of the beam equation. The corresponding Noether integrals of the reduced Euler–Lagrange equations are deduced which remarkably allows for three families of new exact solutions of the static beam equation. These are shown to contain all the previous solutions obtained from the standard Lie analysis and more.