Power Law Nonlinearity Research Articles

Diverse general solitary wave solutions and conserved currents of a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science

This article presents an analytical investigation performed on a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science. To start with, achieving diverse solitary wave solutions to the generalized power‐law model involves using wave transformation, which reduces the model to a nonlinear ordinary differential equation. A direct integration approach is adopted to construct solutions in the beginning. This brings the emergence of interesting solutions like non‐topological solitons, trigonometric functions, exponential functions, elliptic functions, and Weierstrass functions in general structures. Besides, in a bid to secure more general exact solutions to the model, one adopts the extended Jacobi elliptic function expansion technique (for some specific cases of ). Thus, various cnoidal, snoidal, and dnoidal wave solutions to the understudied model are attained. The copolar trio explicated in a tabular form reveals that these solutions can be relapsed to various hyperbolic and trigonometric functions under certain criteria. Additionally, diverse graphical exhibitions of the dynamical attributes of the gained results are presented in a bid to have a sound understanding of the physical phenomena of the underlying model. Later, one gives the conserved vectors of the aforementioned equation by employing the standard multiplier approach.

Nonlinear power-law creep of cell cortex: A minimal model

Experiments have revealed that biological cells exhibit a universal power-law rheology, but the underlying mechanisms remain elusive. Here, we present a minimal model to explain the power-law creep of cell cortex, which is abstracted as chains of crosslinkers with random binding energies. Using this model, we show that when both the load and chain length are small, the logarithm of both the strain and time scales with the fraction of unbound crosslinkers, leading to power-law creep with a constant exponent, as observed in many experiments. Increasing the load alters the latter relationship between time and unbinding fraction, and thus, increases the power-law exponent, explaining the stress-induced nonlinearity in some experiments. Increasing the chain length alters this relationship as well, and as a result, the exponent grows proportionally with the chain length, explaining the crosslinker-density-induced nonlinearity in other experiments. This work provides a mesoscopic explanation for the linear and nonlinear power-law creep of cell cortex and may serve as a basis for understanding the cytoskeletal mechanics.

Modulational instability, generation, and evolution of rogue waves in the generalized fractional nonlinear Schrödinger equations with power-law nonlinearity and rational potentials.

In this paper, we investigate several properties of the modulational instability (MI) and rogue waves (RWs) within the framework of the generalized fractional nonlinear Schrödinger (FNLS) equations with rational potentials. We derive the dispersion relation for a continuous wave (CW), elucidating the relationship between the wavenumber and the instability growth rate of the CW solution in the absence of potentials. This relationship is primarily influenced by the power parameter σ, the Lévy index α, and the nonlinear coefficient g. Our theoretical findings are corroborated by numerical simulations, which demonstrate that MI occurs in the focusing context. Furthermore, we study the RW generations in both cubic and quintic FNLS equations with two types of time-dependent rational potentials, which make both cubic and quintic NLS equations support the exact RW solutions. Specifically, we show that the introduction of these two potentials allows for the excitations of controllable RWs in the defocusing regime. When these two potentials become the time-independent cases such that the stable W-shaped solitons with non-zero backgrounds are generated in these cubic and quintic FNLS equations. Moreover, we consider the excitations of higher-order RWs and investigate the conditions necessary for their generations. Our analysis reveals the intricate interplay between the system parameters and the potential configurations, offering insights into the mechanisms that facilitate the emergence of higher-order RWs. Finally, we find the separated controllable multi-RWs in the defocusing cubic FNLS equation with time-dependent multi-potentials. This comprehensive study not only enhances our understanding of MI and RWs in the fractional nonlinear wave systems, but also paves the way for future research in related nonlinear wave phenomena.

Local structure-preserving algorithms for the nonlinear Schrödinger equation with power law nonlinearity

This paper introduces three local structure-preserving algorithms for the one-dimensional nonlinear Schrödinger equation with power law nonlinearity, comprising two local energy-conserving algorithms and one local momentum-conserving algorithm. Additionally, we extend these local conservation algorithms to achieve global conservation under periodic boundary conditions. Theoretical analyses confirm the conservation properties of these algorithms. In numerical experiments, we validate the advantages of these algorithms in maintaining long-term energy or momentum conservation by comparing them with a multi-symplectic Preissman algorithm.

Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity

Abstract In this article, we consider the global and local well-posedness of the mild solutions to the Cauchy problem of fractional drift diffusion system with higher-order nonlinearity. The main difficulty comes from the higher-order nonlinearity. Instead of the convention that people always focus on the properties of the solution in critical spaces, here we are interested in non-critical spaces such as supercritical Sobolev spaces and subcritical Lebesgue spaces. For the initial data in these non-critical spaces, using the properties of fractional heat semigroup and the classical Hardy-Littlewood-Sobolev inequality, we obtain the existence and uniqueness of the mild solution, together with the decaying rate estimates in terms of time variable.

Extraction of newly soliton wave structure of generalized stochastic NLSE with standard Brownian motion, quintuple power law of nonlinearity and nonlinear chromatic dispersion

Stochastic optical solitons are a fascinating phenomenon in nonlinear optics where soliton-like behavior emerges in systems affected by stochastic noise. This study investigates the influence of Brownian motion on wave propagation in optical fibers. The propagation is modeled using a stochastic nonlinear Schrödinger equation incorporating quintuple power-law nonlinearity and nonlinear chromatic dispersion. To explore this, the improved modified extended tanh (IMET) scheme, leveraging the extended Riccati equation, is employed. This technique facilitates the extraction of various stochastic solutions, including bright, dark, and singular solitons. Furthermore, solutions in the shapes of exponential, singular periodic, and Weierstrass elliptic forms are investigated. The study looks at how the strength of noise impacts various solutions, and Matlab software is used to create 2D and 3D graphs that show the results. It has been noted that when noise intensity rises, signal level falls and surface flattens.

Lie group analysis and conserved vectors of a generalized (3+1)-dimensional Kadomtsev- Petviashvili Benjamin-Bona-Mahony equation with power law nonlinearity

In this paper, our aim is to compute exact solutions for the generalized (3+1)-dimensional KadomtsevPetviashvili Benjamin-Bona–Mahony (gnKP-BBM) equation by invoking an effective method, namely the Lie symmetry technique. Firstly, we derive the infinitesimals and write down the Lie symmetries. Using these symmetries the gnKP-BBM equation is reduced to various nonlinear ordinary differential equations (NLODEs). Thereafter, solutions of the NLODEs are derived by using the Jacobi elliptic cosine method, the (G 0 /G)-expansion method, the simplest equation technique and Kudryashov’s method. Conclusively, we derive conservation laws of the gnKP-BBM equation by using the Ibragimov’s method.

THE NUMERICAL APPROXIMATION OF STATIONARY WAVE SOLUTIONS FOR TWO-COMPONENT SYSTEM OF NONLINEAR SCHRÖDINGER EQUATIONS BY USING GENERALIZATION PETVIASHVILI METHOD

The Petviashvili method is a numerical method for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: , where is a positive definite and self-adjoint operator and is constant. Due to the case being a system of solitary nonlinear wave equations, we generalize the Petviashvili method. We apply this generalized method for a two-component system of Nonlinear Schr dinger Equations (NLSE) for 2-D.

Optical solitons and modulation instability analysis for the generalized resonant dispersive nonlinear Schrödinger equation with dual power-law nonlinearity

Optical solitons and modulation instability analysis for the generalized resonant dispersive nonlinear Schrödinger equation with dual power-law nonlinearity

Dark, bright and singular optical solitons for Biswas–Milovic equation with Kerr and dual power law nonlinearities

This paper attains cubic-quartic optical solitons of the Biswas–Milovic equation with dual power law and Kerr law nonlinearities. Dark, bright, singular and combo-singular solitons solutions are obtained with the F-expansion method. The solutions gained in our this work are newer than the obtained solutions using different method before this work in the literature. In order to better convey the physical significance of the solutions, we also include 2D and 3D graphics. The computational software MATHEMATICA is the helping tool for visualizing and analyzing the derived solutions.

Exact Periodic Wave Solutions for the Perturbed Boussinesq Equation with Power Law Nonlinearity

In this paper, exact periodic wave solutions for the perturbed Boussinesq equation with power law nonlinearity are obtained for different nonlinear strengths n. When n=1, the periodic traveling wave solutions can be found by the definition of the Jacobian elliptic function. When n≥1, we construct a transformation to solve for the power law nonlinearity, and the periodic traveling wave solutions can be obtained by applying the extended trial equation method. In addition, we consider the limiting case where the periodicity of the periodic traveling wave solutions vanishes, and we obtain the soliton solution for n=1. Numerical simulations show the periodicity of the solution for the perturbed Boussinesq equation.

Assorted Spatial Optical Dynamics of a Generalized Fractional Quadruple Nematic Liquid Crystal System in Non-Local Media

Nematicons upgrade the recognition of light localization in the reorientation of non-local media. the current research employs a powerful integral scheme using a different procedure, namely, the modified simple equation method (MSEM), to analyze nematicons in liquid crystals from the controlling model. The expanded MSEM is investigated to enlarge the applicability of the standard one. The suggested expansion depends on merging the MSEM and the ansatz method. The new generalized nonlinear n-times quadruple power law is included. With the aid of the symbolic computational package Mathematica, new explicit complex hyperbolic, periodic, and more exact spatial soliton solutions are derived. Moreover, the related existence constraints are obtained. To show the dynamical properties of some of the obtained nematicons, three-dimensional profiles with corresponding contours are depicted with the choice of appropriate values of arbitrary parameters. The fractional impacts in various applicable senses are analyzed to investigate the generality of the considered model.

New optical soliton solutions to the Biswas–Milovic equations with power law and parabolic law nonlinearity using the Sardar-subequation method

New optical soliton solutions to the Biswas–Milovic equations with power law and parabolic law nonlinearity using the Sardar-subequation method

New optical soliton solutions to nonlinear Schrödinger equation with fractional temporal evolution having parabolic and dual power law nonlinearities

New optical soliton solutions to nonlinear Schrödinger equation with fractional temporal evolution having parabolic and dual power law nonlinearities

On the nonlinearity of the catchment instantaneous response function: Insights obtained from dynamic wave modeling

In the unit hydrograph theory, the catchment response function f to a unit rainfall is described invariant. However, real catchment response is nonlinear in that f varies with rainfall excess rate ie. In particular, the peak f(tp) and peak time tp of f are known to follow power functional relationships with ie. We investigated two fundamental knowledge gaps in the emergence of the nonlinearity: (1) the role of hydrodynamics and (2) the linkage with the linear convolution principle. Employing a sophisticated hydrodynamic model of the tRIBS-FEaST, the power-law nonlinearity was captured in simulations for a series of catchments. We found further that the time of concentration decreases with ie, again following a power law. Nevertheless, we found significant limitations in f obtained by deconvoluting S-hydrographs, as it fails to capture the known tendency of the growing skewness with ie. Further, the hydrographs reproduced with the linear convolution, even the nonlinearity is seen in f, deviates from the continuous simulation results. Alternatively, a series of f was directly simulated as hydrographs from the hydrodynamic model, initiated with varying initial water depth ho all over the catchment, but no additional water or rainfall imposed. This alternative method successfully reproduces not only the nonlinear peak variation but also the skewness variation with ho, which is attributed to the relaxation of the linear convolution framework.

Estimation of fracture behavior of CFRP/CFRP adhesively bonded joints under mixed-mode conditions using a cohesive zone model

Finite element (FE) simulations of adhesively bonded composite joints were conducted using the cohesive zone model (CZM). To clarify the effect of fracture criteria on the accuracy of the CZM simulation, two types of mixed-mode criteria including the ­linear summation model of Modes I and II energy release rates, G I / G IC + G II / G IIC = 1 , and the nonlinear power-law, ( G I / G IC ) n + ( G II / G IIC ) n = 1 , were employed. A CZM formulation using the power-law fracture criterion was developed in this study. The load–displacement curves calculated by the two models were compared with the experimental results obtained from mixed-mode Fernlund–Spelt-type double cantilever beam tests, and the Mode II end-notched flexure (ENF) tests. Calculations using the nonlinear fracture criterion were closer to the experimental results. Consequently, it is concluded that nonlinear fracture modeling is of vital importance for accurate strength analysis of mixed-mode adhesive joints.

Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization

This study extensively explores the Biswas–Milovic equation (BME) with Kerr and power law nonlinearity to extract the unique characteristics of optical soliton solutions. These optical soliton solutions have different applications in the field of precision in optical switching, applications in waveguide design, exploration of nonlinear optical effects, imaging precision, reduced intensity fluctuations, suitability for optical signal processing in optical physics, etc. Through the powerful (G′/G, 1/G)-expansion analytical method, a variety of soliton solutions are expressed in three distinct forms: trigonometric, hyperbolic, and rational expressions. Rigorous validation using Mathematica software ensures precision, while dynamic visual representations vividly portray various soliton patterns such as kink, anti-kink, singular soliton, hyperbolic, dark soliton, and periodic bright soliton solutions. Indeed, a sensitivity analysis was conducted to assess how changes in parameters affect the exact solutions, aiding in the understanding of system behavior and informing decision-making, especially in accurately designing or analyzing real-world optical phenomena. This investigation reveals the significant influence of parameters λ, τ, c, B, and Κ on the precise solutions in Kerr and power law nonlinearities within the BME. Notably, parameter λ exhibits consistently high sensitivity across all scenarios, while parameters τ and c demonstrate pronounced sensitivity in scenario III. The outcomes derived from this method are distinctive and carry significant implications for the dynamics of optical fibers and wave phenomena across various optical systems.

A Couple of Fresh New Perspectives on the Concatenation Model with Power-Law of Self-Phase Modulation

In this paper, we have secured new optical solitons for the concatenation model with power-law nonlinearity. The traveling wave hypothesis serves as the starting point. To retrieve optical soliton solutions, we have implemented two powerful techniques into the model: the Sardar Sub-Equation Method (SSEM) and the Tanh-Coth method. For power-law nonlinearity, we derived through the balancing principle that solitons would exist for different values of the power-law parameter. Therefore, we have secured a large variety of new soliton solutions for the model. This paper derives dark, bright, and singular soliton solutions for the value of n, as the first case was already covered in a previous report dedicated to addressing the model with Kerr law nonlinearity. Lastly, all the parametric existence conditions of the solitons and all solutions have been constructed.

Local existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equation

In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the p-generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying the higher dimensional discrete convolution operation for several functions:c×⋯×c︸p(total distance):=∑♣1,⋯,♣p∈Zν♣1+⋯+♣p=total distance∏j=1pc(♣j). In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for p=2[11], and the combinatorics become harder for larger values of p. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, p=3. Next, we prove existence and uniqueness in the general case p≥2, which then covers the remaining cases p≥4. As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general p.

Investigating dispersive optical solitons with the generalized stochastic perturbed Schrödinger–Hirota equation incorporating power-law nonlinearity and multiplicative white noise

This article aims to introduce the generalized stochastic perturbed Schrödinger–Hirota equation, which incorporates multiplicative white noise in the Itô sense. The study focuses on investigating the stochastic optical soliton solutions related to the governing model to understand the characteristics and properties of these soliton solutions. We can achieve this by utilizing the enhanced Kudryashov method and the enhanced direct algebraic approach. The study stands out for its innovative approach to examining the effects of multiplicative white noise on stochastic optical soliton solutions through the lens of the generalized stochastic perturbed Schrödinger–Hirota equation. Utilizing the enhanced Kudryashov method alongside the novel enhanced direct algebraic method, this research effectively identifies and delineates a range of soliton solution types, such as bright, dark, singular, and straddled solitons. This thorough investigation not only underscores the efficacy and efficiency of these methods but also sheds light on the impact of noise on the structural and dynamic properties of soliton solutions, supported by illustrative visualizations. A notable finding is that all solutions derived using the enhanced direct algebraic method feature a nonzero coefficient for the highest-order term of the auxiliary equation. The diversity of solutions obtained through these approaches includes bright, dark, singular, and straddled solitons. Additionally, the study reveals Weierstrass and Jacobi elliptic doubly periodic solutions, which can transition into various soliton types under specific parameter conditions, offering a broader understanding of soliton behavior and properties.