International Journal of Modern Physics B

The nonlinear fractional differential equations (FDEs) are composed by mathematical modeling through nonlinear corporeal structures. The study of these kinds of models has an energetic position in different fields of applied sciences. In this study, we observe the dynamical behavior of nonlinear traveling waves for the $M$-fractional $(3+1)$-dimensional Wazwaz–Benjamin–Bona–Mohany (WBBM) equation. Novel exact traveling wave solutions in the form of trigonometric, hyperbolic and rational functions are derived using $(1/G^{\prime })$, modified $(G^{\prime }/G^2)$ and new extended direct algebraic methods with the help of symbolic soft computation. We guarantee that all the obtained results are new and verified the main equation. To promote the essential propagated features, some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the precise values to the parameters under the constrain conditions, and this provides useful information about the dynamical behavior. Further, bifurcation behavior of nonlinear traveling waves of the proposed equation is studied with the help of bifurcation theory of planar dynamical systems. It is also observed that the proposed equation support the nonlinear solitary wave, periodic wave, kink and antikink waves and most important supernonlinear periodic wave.

This investigation concerns the phase agreement (PA) for identical Kuramoto oscillators and frequency synchronization (FS) for nonidentical oscillators within fixed-time (FxT) in noisy environments. Comparing with the conventional FxT controllers, this investigation picks the smooth control protocol for stripping the signum function out, which could improve the synchronizing performance of networked systems. Furthermore, due to the ubiquity of stochastic perturbation, this investigation takes the noise into consideration. Additionally, the synchronizing criteria are deduced for realizing the FxT phase-FS in probability, and meanwhile the time estimation (TE) of realizing synchronization is given. At last, simulation results not only validate the correction of theoretical analysis, but also indicate that PA is more robust to stochastic disturbance, whereas FS of oscillators is relatively sensitive.

In this paper, we study the role of degeneracy in the Hubbard model featuring infinite-range Coulomb coupling. We put forward its exact solution for any dimension alongside analytical expressions for the one-dimensional (1D) lattice, thus generalizing the solution proposed by Hatsugai and Kohmoto for a single orbital. The ground-state phase diagram is also addressed against the particle density, interaction strength, and orbital number. We show that increasing the orbital number and/or the Hund’s hole coupling favors the phases associated to aligned spins within orbitals. Standard thermodynamic quantities in the grand canonical ensemble are further evaluated in 1D.

Financial fraud is more likely to spread and produce serious and adverse results through social networks. This study investigates four protection strategies: the uniform protection strategy, the random protection strategy, the targeted protection strategy, and the acquaintance protection strategy based on the potential-investor-divestor (PID) model. The simulation results show that the targeted protection strategy is the best solution for both ER and BA networks. The random protection strategy is the least efficient solution, as it requires spreading a large number of anti-fraud messages to achieve a relatively good performance. The acquaintance protection strategy performs closely to the targeted protection strategy in terms of social dynamics. However, the uniform protection strategy is better than the acquaintance protection strategy, as it involves fewer victims when it collapses. This study suggests that the regulators should protect investors from financial fraud collapses by promoting the financial literacy education and regulating the behaviors of influential people.

In this work, Lie symmetry analysis method is utilized to find the complex soliton solutions of the perturbed Fokas–Lenells equation. In this direction, first of all, we obtained the infinitesimals of the Fokas–Lenells equation with the help of the Lie symmetry method. After that, we reduced the Fokas–Lenells equation into the highly nonlinear system of an ordinary differential equation. Consequently, with the application of suitable back transformation, complex solitons are formulated for the Fokas–Lenells equation in trigonometric, hyperbolic and exponential functions. Finally, in this work, conditions of stability and instability are discussed with the aid of baseband modulation instability.

The main purpose of this work is to study the steady incompressible second-grade micropolar fluid flow over a nonlinear vertical stretching Riga sheet. Velocity slip and zero mass flux are considered at the solid surface of Riga shape such that the friction of nanoparticle maintains itself with strong retardation. The influence of Lorentz forces produced by the Riga plate is an important aspect of the study. The influences of thermophoresis and Brownian motion under the heat generation and e bouncy forces are studied on the nonlinear vertical Riga sheet. The mathematical model is developed under the flow assumptions. The mathematical model in terms of partial differential equations is formed by implementing the boundary layer approximations. The partial differential equations are further reduced to ordinary differential equations by means of suitable transformations. The ordinary differential equations are solved through the numerical procedure. The variations in the horizontal movement of nanofluid, thermal distribution and concentration distribution of the nanoparticle have been noted for different fluid parameters. The values of velocity profile and temperature profile are larger in the case of injection ($S<0)$ as compared to suction ($S>0$). The values of concentration distribution are smaller in the case of injection ($S<0)$ as compared to suction ($S>0)$. The validation of this analysis with decay literature is provided in the form of tables.

This paper is concerned with the rotation-two-component Camassa–Holm (R2CH) system, which is a model for equatorial water waves under the influence of the Coriolis force. The system includes the Dullin–Gottwald–Holm equation, the standard two-component integrable Camassa–Holm (CH) system and the CH equation as special cases. We aim to explore whether the R2CH system admits some generalized peakon weak solutions in the sense of distribution. The exact weak solutions in a particular form are derived by the ansatz method and they are proven to be the generalized peakon weak solutions. These solutions may help to explain the wave-breaking phenomenon in the wave motion to some extent. The method proposed in this paper is effective, which might be applied to study other nonlinear wave equations directly and even to be extend to search for fractal solitary wave.

This investigation includes a three-dimensional Darcy–Forchheimer flow model and the heat transfer phenomenon of H₂O-CNTs nanofluid for a two-way stretchable surface. Xue’s proposed thermal conductivity model is employed. The numerical analysis scheme is applied to solve the transformed PDEs. The outline of velocities, temperature, surface drag forces and Nusselt number against relevant variables are portrayed. From this study, it has been noted that with an increase in Eckert numbers along both directions, two patterns were obtained for temperature curves, the initial temperature outlines increased and after that they decreased. Moreover, the width of the thermal boundary layer for H₂O-MWCNT nanofluid was more compared to H₂O-SWCNT nanofluid. To validate the existing code, numerical outcomes were compared to the earlier published data.

In this paper, the local discontinuous Galerkin method is used to analyze numerical solutions for nonlinear Allen–Cahn equations with nonperiodic boundary conditions. To begin with, the spatial variables are discretized to generate a semidiscrete method of lines scheme. This yields an ordinary differential equation system in the temporal variable, which is then solved using the higher-order total variation diminishing Runge–Kutta method. A comparison of the generated numerical results to the exact results for various test problems using different tables and figures provides insight into the effectiveness and accuracy of the proposed method. The numerical results confirm that the proposed method is an effective numerical scheme for solving the Allen–Cahn equation since the obtained solutions are extremely close to the exact solutions while exhibiting substantially less error.

This study aims at finding the linear theory for the onset of ferromagnetic convective flow in a heterogeneous Brinkman porous layer with uniformly distributed internal heat source in the presence of vertical magnetic field. The resulting critical values are obtained numerically using the Galerkin technique for isothermal/insulated rigid-ferromagnetic boundaries for different forms of vertical heterogeneity permeability function $F(z)$. The results converge for six terms in the Galerkin expansion. The effect of types of $F(z)$ and $N_s$ is found to either delay or speedup the flow of the ferrofluids. The stability of the system for the model $F4$ is more stable and least stable for the model $F1$ in the presence of $N_s$. For different forms of $F(z)$, the results show that the critical Rayleigh number increases with increasing $Da$, while decreases with increasing $R_m$ and $M_3$. The values of $a_c$ increase with $R_m$, but they decrease with increasing $M_3$ and $Da$. Besides, isothermal boundaries are found to be more stabilizing when compared to insulated boundaries.