Narrow Results

For a porous media equation, the existence of uncountably infinite many global monotonic kink wave solutions with bounded supports is proved. By using the method of planar dynamical systems, the dynamical behavior of the corresponding traveling wave system is discussed. A few exact explicit parametric representations of the kink wave solution families are given.

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales.

In this paper, we discuss various analytical solutions of the planner geometry with anisotropic fluid as well as vanishing complexity factor condition in the framework of $f(𝔾)$ theory. The matching conditions are employed at the interior and exterior hypersurfaces. The connection between the matter parameters and the Weyl tensor has been constructed. For dissipative and non-dissipative structures, we examine four different conformal Killing vector possibilities, while some possibilities adhere to the ${{𝒴}}_{TF}=0$ condition along with the matching conditions, where ${{𝒴}}_{TF}$ indicates the trace free part of the electric component of the Riemann tensor. Several theoretical models are provided to show how a non-static system evolves.

An important manifestation of the integrability of nonlinear mathematical and physical models is their solvability through inverse scattering transform (IST). The problem to be investigated in this paper is a system of fifth-order nonlinear evolution equations (NLEEs) with variable coefficients related to time-varying spectral problems. The expected result is to derive the system of fifth-order NLEEs, obtain its exact solutions, verify its integrability in the sense IST solvability, and reveal some novel local structures of the obtained solutions. First, the system of fifth-order NLEEs is derived. Then, by combining the IST with a time-varying spectral parameter, the associated scattering data are determined for the reconstruction of potentials. Using the determined scattering data, exact solutions are ultimately obtained. Meanwhile, some local structures with new features are analyzed for two pairs of special solutions, including kink solitons, wide/narrow top bell solitons, and small-scale peaks in the time variable direction, as well as sine/cosine-like fluctuations in the spatial variable direction. This paper not only demonstrates that equipping the Ablowitz–Kaup–Newell–Segur (AKNS) problems with appropriate time-varying spectra can be used to derive some other inverse scattering integrable systems of NLEEs with variable coefficients, but also graphically illustrates the regulatory effect of time-varying spectral parameter on local solution structures.

This work aims to investigate the self-bound anisotropic solution for spherical objects within the $f({𝒬})$ gravity, where $Q$ is called a nonmetric scalar. Initially, the equation of motion for gravity theory is obtained by using a linear representation of the function $f({𝒬})$ as $f({𝒬})={𝜖}_{1}Q+{𝜖}_2$, here parameters ${𝜖}_1$ and ${𝜖}_2$ are used. Subsequently, the acquired system of differential equations was solved by including a radial metric component in conjunction with the specific ansatz of the anisotropy factor. The values of the constants required for the solution were established by using the Schwarzschild (Anti-) de Sitter exterior solution. In order to assess the physical acceptability of a solution, it is necessary to examine several physical conditions, including the behavior of pressure, density, adiabatic index, and equilibrium conditions, for different values of the parameter ‘${𝜖}_1$’ inside the star system. These requirements are visually represented by graphical analysis. The current solution meets all the physical requirements, indicating that it is a suitable model of a compact stellar structure.

The prime aim and essence of this study are to present a closed-form solution of the unidirectional velocity field for thin film flow generated by a third-grade liquid moving over a fixed or movable inclined plane in the presence of partial slip boundary condition. Consideration of partial slip makes this problem different from other published research works. Here lies the novelty of this study. The nondimensional, unidirectional velocity profiles rely on the material parameter of third-grade fluid, slip parameter and initial velocity of the movable plane. Study discloses that the fluid’s velocity decelerates with raising the material property of grade-3 fluid and it accelerates with the slip parameter and initial velocity of the inclined plane. The outcomes of this study are deliberated physically at length.

We apply Painlevé test to the most general variable coefficient nonlinear Schrödinger (VCNLS) equations as an attempt to identify integrable classes and compare our results versus those obtained by the use of other tools like group-theoretical approach and the Lax pairs technique of the soliton theory. We presented some exact solutions based on point transformations relating analytic solutions of VCNLS equations for specific choices of the coefficients to those of the standard integrable NLS equation.

Under investigation in this paper is a more general discrete $2\times 2$ matrix spectral problem. Starting from this spectral problem, the positive and negative integrable lattice hierarchies are constructed based on the Tu scheme, then by considering linear combination of the positive and negative lattice hierarchies, we give a more general integrable lattice hierarchy, which can reduce to the well-known Ablowitz–Ladik lattice and the discrete modified Korteweg–de Vries (mKdV) equation. In particular, we obtain some local and nonlocal integrable lattice equations, including reverse-space discrete mKdV equation, reverse-space complex discrete mKdV equation, higher-order discrete mKdV equation, higher-order complex discrete mKdV equation, higher-order reverse-space discrete mKdV equation and higher-order reverse-space complex discrete mKdV equation. In additional, infinitely many conservation laws and Darboux transformation (DT) for the first non-trivial system in the hierarchy are established with the help of its Lax pair. The exact solutions of the system are generated by applying the obtained DT. The results in this paper might be helpful for understanding some physical phenomena.

The projective Riccati equations method is extended to find some novel exact solutions of a (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation. Applying the extended method and symbolic computation, six families of exact analytical solutions for this NLS equation are reported, which include some new and more general exact soliton-like solutions, trigonometric function forms solutions and rational forms solutions.

An exact solution is presented for mixing two or more different types of Random Deposition (RD) nonequilibrium surface growth processes. The depositions may be made in a sequential mode, or in a random mode by first randomly choosing the lattice site for deposition. The results hold in all dimensions d. Simulations are presented in d = 1 for comparison. Furthermore, a mean-field type of approach to mixing RD with other surface growth processes is tested against the exact solution.

This study explores heat and mass transport in natural convection of Casson fluid in a vertical annulus via porous medium. Impacts of thermal radiation, heat source and chemical reaction are taken into consideration. The equations representing the model reduced into nondimensional ordinary differential equations under adequate transformations are solved analytically. Closed form solutions are obtained for the problem in terms of Bessel’s functions. Influences of various arising parameters such as porous medium parameter, heat generation, thermal radiation, thermal Grashof number, solutal Grashof number, etc. on flow, temperature and concentration fields are exhibited by graphs and discussed. Also, we have solved the problem numerically on MATLAB software employing the bvp4c technique along with shooting technique. The exact and numerical solutions compared found a good match. Moreover, the effects of numerous parameters on quantities of physical importance such as skin-friction coefficient, Nusselt number and Sherwood number are also portrayed and discussed. Heat exchangers, energy storage systems such as batteries and inverters, thermal storage and thermal protection systems are some examples of applications of the study.

We obtain a large class of solvable potentials and the corresponding exact solutions of the time-dependent Schrödinger equation with time-dependent mass by transforming it into an equation of telegraphy.

Extending the method presented in our previous paper,¹² we map the time-dependent Schrödinger equation (TDSE) with time-dependent mass on a stationary Schrödinger equation for a nonconstant potential. On solving the latter, we can thus generate a large class of exact solutions of the original TDSE. Several examples are given, including potentials of power-law and modified Pöschl–Teller type.

We show that the time-dependent Schrödinger equation (TDSE) for a potential of the form V(x,t)=A(t)x²+B(t)x+C(t) and time-dependent mass can be transformed into the same TDSE with constant mass. We obtain an explicit formula relating solutions of the TDSE for time-dependent mass and for constant mass to each other.

A class of exact static solution around a global monopole resulting from the breaking of a global SO(3) symmetry is obtained in the context of Lyra geometry.

Our solution is shown to possess an interesting feature like "wormholes" spacetime. It has been shown that the global monopole exerts no gravitational force on surrounding nonrelativistic matter.

We study the non-relativistic Schrödinger equation for a free quantum particle constrained to the surface of a degenerate torus, parametrized by its polar and azimuthal angle. On restricting to wave functions that depend on the polar angle only, the Schrödinger equation becomes exactly-solvable. We compute its physical solutions (continuous, normalizable and 2π-periodic) and the associated energies in closed form.

The unjustly neglected method of exactly solving generalized electroweak models — with an original spontaneous symmetry breaking mechanism based on the gauge group SU(n)_L⊗U(1)_Y — is applied here to a particular class of chiral 3-3-1 models. This procedure enables us, without resorting to any approximation, to express the boson mass spectrum and charges of the particles involved therein as a straightforward consequence of both a proper parametrization of the Higgs sector and a new generalized Weinberg transformation. We prove that the resulting values can accommodate the experimental ones just by tuning a sole parameter. Furthermore, if we take into consideration both the left-handed and right-handed components of the neutrino (included in a lepton triplet along with their corresponding left-handed charged partner), then we are in the position to propose an original method for the neutrino to acquire a very small but nonzero mass without spoiling the previously achieved results in the exact solution of the model. In order to be compatible with the existing phenomenological data, the range of that sole parameter imposes a large order of magnitude for the vev〈ϕ〉~10⁴ TeV in our method. Consequently, the new bosons of the model have to be very massive.

The mass splittings for the Majorana neutrinos in the exact solution of a particular 3-3-1 gauge model are computed in detail. Since both sin²θ₁₃≃0 and the mass splitting ratio r_Δ≃0.033 are taken into account as phenomenological evidence, the analytical calculations in the leading order seem to predict an inverted mass hierarchy and a matrix with a texture based on a very close approximation to the bi-maximal mixing. The resulting formulas for the mass squared differences can naturally accommodate the available data if the unique free parameter a of the model gets very small values ~ 10^-15. Consequently, the smallness of the parameter requires (according to our method) a large breaking scale 〈ϕ〉 ~ 10⁶–10⁷ TeV. Hence, the results concerning the neutrino mass splittings lead to a more precise tuning in the exact solution of the 3-3-1 model and enable it — at the same time — to recover all the Standard Model phenomenology and predict the mass spectrum of the new gauge bosons Z′, X, Y in accordance with the actual data. The minimal absolute mass in the neutrino sector is also obtained, m₀≃0.0035 eV, in the case of our suitable approximation for the bi-maximal mixing.

The exact solutions of the one-dimensional Klein–Gordon equation for the Rosen–Morse type potential with equal scalar and vector potentials are presented. First, we briefly review Nikiforov–Uvarov mathematical method. Using this method, wave functions and corresponding exact energy equation are obtained for the s-wave bound state. It has been shown that the results for Rosen–Morse type potentials reduce to the standard Rosen–Morse well and Eckart potentials in the special case. The PT-symmetry for these potentials is also considered.

In this paper we present the analytical solutions of the one-dimensional Dirac equation for the Scarf-type potential with equal scalar and vector potentials. Using Nikiforov–Uvarov mathematical method, spinor wave function and the corresponding exact energy equation are obtained for the s-wave bound state. It has been shown that the results for this potential reduce to the well-known potentials in the special cases.