Narrow Results

This study investigates the dynamical attitude of a nonlinear elastic circular rod’s longitudinal oscillation with lateral inertia and finite radius. This model was derived in 1986 by Wei and Gui-tong with a fourth-order nonlinear mixed derivative. The axial symmetry of this model has been thought through by using cylindrical coordinates. Furthermore, the strain and kinetic energy in the length unit of the rod have been determined. Two recent computational (extended Fan-expansion (EFE) and generalized rational (GR)) techniques are employed to construct some novel solitary wave solutions. The soliton wave solutions are obtained using Mathematica 13 software and are given with the distinct physical properties of trigonometric, hyperbolic and rational solution species. The stability of the investigated model and the obtained solutions through the suggested two analytical schemes are tested. Putting different values of the parameters explains these solutions through some numerical simulations in two-dimensional, three-dimensional and contour plots.

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution’s quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated.

The morphology of a shallow-water wave is affected by the unsmooth boundary, while its peak is rarely changed. This phenomenon cannot be explained by a differential model. This paper adopts a fractal modification of the Boussinesq equation, and its traveling solitary solution is studied through its fractal variational principle, the results reveal the basic properties of solitary waves in fractal space.

In this study, we construct several wave solutions to the nonlinear fractional Hirota–Maccari equation with a truncated M-fractional derivative via the extended sinh-Gordon equation expansion method. The constraint conditions that guarantee the existence of valid solutions are stated. We use suitable values of parameters in plotting the 2- and 3-dimensional graphs of the reported solutions.

This paper discusses bright, dark and singular optical soliton as well as complexiton solutions to the coupled Fokas–Lenells equation (FLE) for birefringent fibers by three integration tools such as $exp(-\varphi (\chi ))$-expansion method, the first integral method and the sine-Gordon expansion method. The existence criterion of these solutions is also given.

A formulation of Darboux transformations is proposed for integrable couplings, based on non-semisimple matrix Lie algebras. Applications to a kind of integrable couplings of the AKNS equations are made, along with an explicit formula for the associated Bäcklund transformation. Exact one-soliton-like solutions are computed for the integrable couplings of the second- and third-order AKNS equations, and a type of reduction is created to generate integrable couplings and their one-soliton-like solutions for the NLS and MKdV equations.

We review recent progress in the computation of leading quantum corrections to the energies of classical solitons with topological structure, including multi-soliton models in one space dimension and string configurations in three space dimensions. Taking advantage of analytic continuation techniques to efficiently organize the calculations, we show how quantum corrections affect the stability of solitons in the Shifman–Voloshin model, stabilize charged electroweak strings coupled to a heavy fermion doublet, and bind Nielsen–Olesen vortices at the classical transition between type I and type II superconductors.

This paper focuses on the numerical investigation of the fractal modification of the (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation. A variational approach based on the two-scale fractal complex transformation and the variational principle is presented for solving this fractal equation. The fractal potential-YTSF equation can be transformed as the original potential-YTSF equation by means of the fractal complex transformation. Some fractal soliton-type solutions and fractal periodic wave solutions are provided by using the variational principle proposed by He, which are not touched in the existing literature. Some remarks about the variational formulation and the wave solutions for the original potential-YTSF equation by Manafian et al. [East Asian J. Appl. Math. 10(3) (2020) 549–565] are also given. Numerical results of the fractal wave solutions with different fractal dimensions and amplitudes are presented to show the propagation behavior.

In this paper, modulational instability (MI) of information via membrane potential is studied analytically and numerically in an improved Izhikevich neural network under electromagnetic induction. By applying the powerful discrete multiple scale expansion method, a spatiotemporal nonlinear amplitude differential-difference equation governing the information dynamics is derived from the generic model. Linear stability of plane impulse wave solution is then performed on the latter and the impact of electromagnetic induction feedback through the memristor couplings is portrayed on the growth rate diagram. From the diagram, it is found that negative memristor coupling parameter decreases the critical amplitude while positive parameter increases the critical amplitude. To support our analytical predictions, numerical simulations are performed and data selected from the unstable zone of MI lead to the formation of localized solitonic energy patterns, related to the energy coding patterns in the nervous system. Furthermore, the sampled time series for membrane potential under the influence of memristor coupling revealed the breakdown of action potential into multiple impulse-wave trains for high parameter values thus confirming an analytical prediction. Our results provide a potential way to manipulate information coding in the brain.

In this scientific work, we explore the solutions of an important mathematical model appearing in the nonlinear elastic inhomogeneous Murnaghan’s rod (NEIMR) or called doubly dispersive equation (DDE) with a useful and powerful analytical technique; which is the extended Kudryashov method with the Bernoulli–Riccati approach. In our research, we have successfully obtained some important solutions of the above-mentioned model such as topological, nontopological, singular solitons, singular periodic and rational solutions. Using the appropriate values of the parameters, we clearly plotted the solutions of the model, 3D, 2D and contour plots. We think that the method used in the study and the results obtained will be an effective motivation tool for many researchers doing research in this field.

Envelope solitons of the Nonlinear Schrödinger equation (NLS) under quantum potential's influence are studied. Corresponding problem is found to be integrable for an arbitrary strength, s ≠ 1, of the quantum potential. For s < 1, the model is equivalent to the usual NLS with rescaled coupling constant, while for s > 1, to the reaction–diffusion system. The last one is related to the anti-de Sitter (AdS) space valued Heisenberg model, realizing a particular gauge fixing condition of the (1+1)-dimensional Jackiw–Teitelboim gravity. For this gravity model, by the Madelung fluid representation we derive the acoustic form of the space–time metric. The space–time points, where dispersion changes the sign, correspond to the event horizon, while the soliton solution to the AdS black hole. Moving with the above bounded velocity, it describes evolution on the one sheet hyperboloid with nontrivial winding number, and creates under collision, the resonance states which we study by the Hirota bilinear method.

Aortic dissection (AD) is the most common catastrophic disease reported at cardiovascular emergency in hospitals. Herein, a tear in the tunica intima results into separation of layers of aortic wall leading to rupture and torrential bleed. Hypoxia and oxidative stress are associated with AD. The release of hypoxia inducible factor (HIF)-$1\alpha$ from the initial flap lesion in the tunica intima is the basis for aneurysmal prone factors. We framed a boundary value problem (BVP) to evaluate homeostatic saturation for oxygen dynamics using steady-state analysis. We prove uniqueness and existence of the solution of the BVP for gas exchange at capillary–tissue interface as a normal physiological function. Failure of homeostatic mechanism establishes hypoxia, a new quasi-steady-state in AD. We model permeation of two-layer fluid comprised of blood and HIF-$1\alpha$ through tunica media as a generalized $(3+1)$-dimensional nonlinear evolution equation and solve it using Lie group of transformations method. We note that the two-layer fluid permeates the tunica media as solitary wave including solitons such as bright soliton, dark soliton, peregrine soliton, topological soliton, kink soliton, breather soliton and multi-soliton complex. Also, we introduce the main result and discuss the implications of soliton solution, using graphic interpretation, to describe the early stage of progression of AD.

In this paper, we study the Ricci–Bourguignon flow of all locally homogenous geometries on closed three-dimensional manifolds. We also consider the evolution of the Yamabe constant under the Ricci–Bourguignon flow. Finally, we prove some results for the Bach-flat shrinking gradient soliton to the Ricci–Bourguignon flow.

In this research, the modified extended tanh-function (METF) and the extended Jacobi elliptic function expansion (EJEFE) techniques are used to investigate the generation and detection of soliton structures in the Hirota–Maccari (HM) model. Consequently, we obtain soliton solutions with advanced structures, including singular bright soliton, dark soliton, periodic waves, breather waves, periodic breather waves, and multiple bright and dark breather waves. In addition, a lump-type breather wave is also included in the presented solutions. Stability analysis of the obtained solutions is addressed by employing the Hamiltonian technique. 3D surfaces and 2D visuals of the outcomes are represented with the help of a computer application. These findings contribute to understanding nonlinear wave phenomena with potential applications in optics, fluid dynamics, and plasma physics.

Soliton solutions which signify long wave parallel to the magnetic fields of Kaup–Newell optical fiber model are discussed in this paper by two different methods. The improved simple equation method (ISEM) and exp$(-\mathrm{\Phi }(\xi ))$-expansion scheme are employed to solve the model to construct the solutions of the model in different cases. The achieved solutions are represented in different and general forms such as logarithmic or exponential function, trigonometric and hyperbolic trigonometric functions, etc. Also, the modulation instability of the model is analyzed which confirms that all obtained exact results are stable. Several solutions from achieved solutions are novel.

In the all-optical domain, the reflective semiconductor optical amplifier (RSOA) creates due attention in the research community because the RSOA has high gain, low-power consumption, high-speed signal processing ability as well as low noise performance. In this work, we have proposed dibit-based OR and NOR gates using RSOA. Dibit-based logic is utilized in this design to get the reduced bit-error problems and high degree of parallelism. To verify the functionality of the aforementioned designs, we have used MATLAB software and also the quality factor (Q), extinction ratio (ER), contrast ratio (CR), and bit-error-rate (BER) have been calculated.

In this paper, the optical soliton and solitary wave solutions of the $(2+1)$-dimensional Mel’nikov equation are investigated using the Kudryashov $R$ function technique. The Kudryashov $R$ function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the $(2+1)$-dimensional Mel’nikov equation have been displayed in 3D, 2D and contour graphs.

In this paper, two important pseudo-parabolic type equations are studied for extracting soliton solutions via the generalized projective riccati equations method. These equations are Oskolkov equation and Oskolkov–Benjamin–Bona–Mahony–Burgers equation. The proposed method extracts dark soliton and singular soliton. Furthermore, the Painlevé test (P-test) has also been employed on pseudo-parabolic type equations for investigating integrability. The proposed equations are proved to be integrable by P-test. The numerical simulations have also been carried out by 3D and 2D graphs of some of the obtained solutions.

Results of numerical study of lump solutions (2D solitons) of a generalised 2D Gardner equation are presented. In the moving coordinate frame the equation is:

We review recent results concerning the interactions of solitary waves for several universal nonlinear dispersive or wave equations. Though using quite different techniques, these results are partly inspired by classical papers based on the inverse scattering theory for integrable models.