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Einstein's equation with a negative cosmological constant admit solutions which are asymptotically anti-de Sitter. We obtain Vaidya-like solutions to include both a null fluid and a string fluid in nonspherical (plane symmetric and cylindrical symmetric) anti-de Sitter space–times. Assuming that string fluid diffuse, we find analytical solutions to Einstein's field equations. Thus we extend the approach proposed by Glass and Krisch to nonspherical space–times with a negative cosmological constant. Some other exacts solutions are also presented and discussed.

In this paper, we study the behavior of static spherically symmetric relativistic model of the strange star SAX J1808.4-3658 by exploring a new exact solution for anisotropic matter distribution. We analyze the comprehensive structure of the space–time within the stellar configuration by using the Einstein field equations amalgamated with quadratic equation of state (EoS). Further, we compare solutions of quadratic EoS model with modified Bose–Einstein condensation EoS and linear EoS models which can be generated by a suitable choice of parameters in quadratic EoS model. Subsequently, we compare the properties of strange star SAX J1808.4-3658 for all the three EoS models with the help of graphical representations.

In this paper, tunneling effect of Cooper pairs in weak-link superconductor structure with multi-junctions under the condition for overstepping the Josephson approximation is discussed. The equations describing the electric current based on the tunneling effect of Cooper pairs in several kinds of weak-link superconductor structures with multi-junctions are obtained under the condition to overstep the Josephson approximation. For both SISISIS and a four junctions ring, when all junctions are in the zero voltage state, the exact solution of the equations is obtained. It is found that, because of the tunneling effect of the Cooper pairs, an alternating current exists which can be expressed by an elliptic function. For a four junctions ring, the relation between the period of the alternating current and flux is pointed out. At last, the condition for overstepping the Josephson approximation is discussed. The result shows that overstepping the Josephson approximation may be possible when the volumes of superconductors are small enough.

In this paper, we derive nonisospectral AKNS equations from the AKNS spectral problem. By using a new Loop algebra , their integrable couplings are constructed. Moreover, bilinear forms of the nonisospectral AKNS equation are given. New exact solutions are obtained through the Hirota method. Some nonisospectral characteristics of the obtained solutions are discussed.

With the help of a mapping approach, a new type of variable separation solution with two arbitrary functions to (2+1)-dimensional Boiti–Leon–Pempinelli system (BLP) is derived. Based on the derived variable separation solution, some single valued and multiple valued localized excitations such as dromions, peakons and foldons, etc. with novel evolutional properties are revealed by introducing appropriate initial conditions in this paper.

With the aid of an extended projective method and a variable separation approach, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for (2+1)-dimensional general Korteweg–de Vries (GKdV) system are derived. Analytical investigation of the (2+1)-dimensional GKdV system shows the existence of abundant stable localized coherent excitations such as dromions, lumps, peakons, compactons and ring soliton solutions as well as rich fractal and chaotic localized patterns in terms of the derived solitary solutions or the variable separation solutions when we consider appropriate boundary conditions and/or initial qualifications.

Based on the one-parameter Lie group theory, we established a modified symmetry reduction method in solving nonlinear variable coefficient equations. Our study shows that the modified method can be applied in solving or reducing various nonlinear variable coefficient equations. In our initial applications, we have successfully obtained some exact solutions to the equations of nonlinear variable coefficients KdV and KP.

Starting from a Painlevé–Bäcklund transformation, an exact variable separation solution with four arbitrary functions for the (2+1)-dimensional generalized Sasa–Satsuma (GSS) system are derived. Based on the derived exact solutions in the paper, some complex wave excitations in the (2+1)-dimensional GSS system and revealed, which describe solitons moving on a periodic wave background. Some interesting evolutional properties for these solitary waves propagating on the periodic wave background are also briefly discussed.

A method for constructing the solutions of nonlinear evolution equations by using the Weierstrass elliptic function and its first-order derivative was presented. This technique was then applied to Burgers and Klein–Gordon equations which showed its efficiency and validality for exactly some solving nonlinear evolution equations.

Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision.

We study the one-dimensional double-exchange model with L localized spins and one mobile electron. We solve the Schrödinger equation analytically and obtain the energies and wavefunctions for all the eigenstates with spin S = (l-1)/2 exactly. As an application, we compute the single-particle Green's function. We show that, for vanishing exchange interactions between localized spins, the single-particle spectrum is entirely incoherent and the lowest band has an infinite band mass, i.e., the single electron is localized due to its interaction with the spin excitations. For nonvanishing exchange interactions between localized spins, the lower edge of the spectrum acquires a dispersion but the spectrum remains incoherent with no well-defined quasiparticle peak.

In this paper, we study a 6-field integrable lattice system, which, in some special cases, can be reduced to the self-dual network equation, the discrete second-order nonlinear Schrödinger equation and the relativistic Volterra lattice equation. With the help of the Lax pair, we construct infinitely many conservation laws and a new Darboux transformation for system. Exact solutions resulting from the obtained Darboux transformation are presented by using a given seed solution. Further, we generate the soliton solutions and plot the figures of one-soliton solutions with properly parameters.

In this paper, a new exact solution of the conformable Gilson–Pickering equation is investigated. It should be noted that some of the individual cases of the Gilson–Pickering equation are the conformable Camassa–Holm, the conformable Fornberg–Whitham, and the conformable Rosenau–Hyman equations. A new version of modified Kudryashov’s method with the help of the mathematical software package is employed to carry out this aim. It is believed that the new modified Kudryashov’s method is well suited, such that it can adapt to a broad range of partial differential equations.

In this research paper, a simple integration scheme is executed to secure new dark and singular soliton solutions for the highly dispersive nonlinear Schrödinger’s equation having Kudryashov’s arbitrary form with generalized nonlocal laws and sextic-power law refractive index.

This paper examines nonlinear partial differential equation (PDE) solutions. Scientists and engineers have struggled to solve nonlinear differential equations. Nonlinear equations arrive in nearly all problems in nature. There are no well-established techniques for solving all nonlinear equations, and efforts have been made to enhance approaches for a specific class of problems. Keeping this in mind, we shall investigate the perturbation method’s efficiency in solving nonlinear PDEs. Several techniques work well for diverse issues. We recognize that there may be several solutions to a given nonlinear issue. Methods include homotropy analysis, tangent hyperbolic function, factorization and trial function. However, some of these strategies do not cover all nonlinear issue solutions. In this paper, we use the perturbation technique to solve the zeroth-order Airy equation and also find the Bessel function in the first-order nonhomogeneous differential equation by using self-similar solutions that appears in modified Korteweg–de Vries (KdV) equation. This approach will be used for nonlinear equations in physics and applied mathematics.

This paper deals with a study on flow of fluid which exhibits the characteristics of both ideal fluids and elastic solid and shows partial elastic recovery. For these types of fluids, Jeffrey six-constant model will be used that illustrates the most striking feature connected with the deformation of a viscoelastic substance and simultaneously displays the fluid-like and solid-like characteristics. The flow of the proposed fluid model will be generated in an inclined tube by sinusoidal wave trains propagation with constant speed along the walls of the tube. The governing equations of the fluid along with energy equation are modeled and simplified by using low Reynolds number and long wavelength assumptions. These equations will be solved by utilizing the homotopy perturbation technique and results of flow will be displayed in graphical form under the effects of Jeffrey model’s parameters.

Heat transfer and entropy generation are crucial considerations in the nuclear industry, where the safe and efficient transfer of heat is essential for the operation of nuclear reactors and other nuclear systems. Casson fluid is a useful tool in the nuclear industry for simulating the flow behavior of nuclear fuels and coolants, and for optimizing the design and operation of nuclear reactors. In view of this, the current investigation deals with the heat and fluid flow of unsteady Casson fluid in a circular pipe under the influence of magnetic field, internal heat generation, entropy generation and porous media. The governing equations have been simplified under suitable assumptions and nondimensional quantities. The simplified dimensionless governing equations have been solved using the method of separation of variables along with Bessel functions. It is concluded from the investigation that the temperature increases with time. The Casson fluid parameter raises the temperature and entropy generation. The temperature, entropy generation and Bejan number are the decreasing functions of the Prandtl number.

The velocity of an unsteady flow of a viscous fluid of the second-grade MHD-type enclosed between two parallel side walls perpendicular to a plate was obtained by applying the integral transformation. The fluid is required to move by the plate, which over time $t=0^{+}$ subjected the fluid to shear stress. The solutions satisfy the given equation as well as the boundary and initial conditions, and they were separated into two types: steady state and transient state. Furthermore, through $h\to \infty$, we are able to recover the results found in the literature for motion across an infinite plate. Graphs depict the effect of the side walls and the time it takes to reach the steady state. The solutions are shown in graphs and discussed physically to examine the impact of different flow parameters. It is found that the fluid velocity decreases with an increasing fractional parameter $\beta$ and second-grade parameter $\alpha$. Also, it is noticed that the fluid velocity decreases with increasing values of Reynolds number and effective permeability. Numerous industrial products, including honey, paints, varnishes, coffee, chocolate and jelly, use this type of fluid flow concept.

To investigate the effects of magnetic fields and variously structured nanoparticles in narrowing, stenosed arteries, an arterial flow model is incorporated. The aim of this study here is to achieve more realistic results by modeling and simulating the arterial blood flow system with the nanoparticles and shape factor of the nanoparticles. The study of blood flow in tapered stenosed arteries with nanoparticles involves understanding the dynamics of blood circulation in vessels having application in drug delivery. Nanoparticles with specific shape factors can enhance imaging modalities like MRI, CT or ultrasound diagnosing tumor. Metallic nanoparticles in various shapes utilizing water as the base fluid have not yet been considered. Imagine a symmetrical radially but axially nonsymmetric constriction for the blood flow. Along with taking into account the regularity in the sequence of distributing the wall shear stress as well as resistance impedance, the study also takes into account a rise in these readings as the stenosis worsens. For speed, resistive impedance, wall shear and shearing forces at the stenosis throat, results have been computed in exact form. For various Cu-water-relevant parameters, the visual results of several types of converging tapering arteries have been assessed.

The exact phase diagram and the ground-state properties of the one-dimensional Hubbard model with arbitrary on-site interaction of electrons are calculated over a wide range of magnetic field and electron concentrations by means of the Bethe-ansatz formalism. The ground-state properties, including the total energy, the average spin (magnetization) and spin (magnetic) susceptibility are investigated for both signs of the interaction strength U/t. The critical behavior near the onset of magnetization and magnetic saturation are also analyzed. At the onset of magnetization and near the magnetic saturation the spin susceptibility χ diverges at all U/t for half-filling case n=1, whereas for n≠1 it is always finite. The reverse susceptibility χ^-1(U) exhibits anomalous hump, which increases with h or n, and shows discontinuity as U/t→±0 at infinitesimal h→0. The analytical results for the ground-state properties in strong and weak interaction limits are in full agreement with our numerical calculations.