The goal of this work is to improve our understanding of the dynamical aspects of the nonlinear Landau-Ginzburg-Higgs (LGH) equation, which offers a theoretical framework for characterizing several phenomena, including the spontaneous breakdown of symmetries and the emergence of superconducting states. The proposed model integrates the concepts of the Higgs mechanism and the Landau-Ginzburg theory when symmetry breaking appears in phase transitions in particle physics or condensed matter systems. The equation is essential for describing the Higgs field and its constituent particles, such as the Higgs boson. New extended direct algebraic approaches and modified F-expansion techniques are used to address this problem. The obtained solutions, which include kink, anti-kink, bright, dark, and periodic solitons, are essential because they shed light on the stability and nonlinear dynamics of field theories that are pertinent to cosmology and condensed matter physics. To advance the essential propagating features, a few obtained solutions are presented as 3D, contour, and 2D graphics by applying certain values to the parameters under the given constraints. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. Additionally, the chaos analysis is carried out to show the quasi-periodic and periodic chaotic patterns. The sensitivity analysis of the studied model is also looked at and presented at different initial conditions. Furthermore, we guarantee that every found solution is precise, effective, and a great addition to the literature of solitary wave theory. To calculate the soliton solutions for nonlinear models in communication engineering and operations research, the methods utilized in this work to derive inclusive and standard solutions are more accessible, effective, and quick.

When this article was published, some mistakes remained undetected. 1. In the first line below equation (7), Instead of has been changed to 2. In Equation (24), instead of there should be 3. Instead of Lemma3.5 (i)Forany wehave. there should be Lemma 3.5 (i) For any we have 4. In Lemma 3.5, instead of where on there should be where on 5. In Lemma 3.5, instead of Takingtheinfimumoverall satisfying, weseethat. there should be Taking the infimum over all satisfying we see that 6. In Lemma 4.3, instead of for any there should be for any 7. Instead of Theorem 5.1 ( [15]). Forany, thereexistsauniquesolution there should be Theorem 5.1 ([15]). For any there exists a unique solution 8. In section “A Splitting lemma”, instead of For, set there should be For set 9. In Lemma A.1, instead of There are there should be There are and 10. In Lemma A.1, instead of Let there should be Let be such that 11. In Reference 5, instead of Berthoumieu, J.: Minimizing travelling waves for the one-dimensional nonlinear schrödinger equation with non-zero condition at infinity, [SPACE] arXiv:2305.17516 (2023) there should be Berthoumieu, J.: Minimizing travelling waves for the one-dimensional nonlinear schrödinger equation with non-zero condition at infinity, arXiv:2305.17516 (2023) 12. In Reference 6, instead of “Béthuel,F.” there should be “Béthuel, F.” 13. In Reference 17, instead of “Gilbarg,D.” there should be “Gilbarg, D.” 14. In Reference 28, instead of “Zhidkov,P.E.” there should be “Zhidkov, P.E.” The Original article has been corrected.

In this paper, an application of the generalized absolute value equation is exploited. Concretely, by using the generalized absolute value equation, we provide a new approach to efficiently obtain the hull of the solution set to interval linear equations with certain inverse positive coefficient interval matrix. Compared with some existed classical methods such as the magnitude method, Krawczyk method, the interval Gauss-Seidel method, and sign accord algorithm, in the implementations, this new approach only requires finding the solutions of two absolute value equations, which only involves the matrix-vector multiplication, and avoids the complex interval-matrix-vector multiplication. To make sure that the proposed method is convergent, some convergence conditions are presented. To confirm the performance of the proposed method, some numerical examples are provided.

In this paper, the asymptotic stability of monotonically increasing kinked soliton solution to the generalized Boussinesq equation with dissipation term is studied by using the the antiderivative method. In order to overcome the difficulty of integral estimate about perturbed solution caused by higher-order derivatives in the equation, several important properties of the nonlinear part for the studied equation and the wave function are first given and proved in this paper. Furthermore, according to the correspondence between the solitary wave solution and the solution trajectory in the plane system, the second-order derivative expression concerning the wave function is given by considering the asymptotic behavior of the solution trajectory at infinity, so that we can successfully obtain the integral estimates of product terms about the second-order derivative of the wave function and the perturbed solution, and then obtain the uniform a priori estimate of studied solitary wave solution with respect to the perturbation, and prove the monotonically increasing kinked soliton solution for the generalized Boussinesq equation with dissipation term is asymptotically stable.

In this study, we have focused on examining the steady motion of a nanofluid characterized by tangent hyperbolic properties as it traverses across a vertically elongating surface. In the current analysis, we take into account the effects of the slip phenomena as well as the influence of thermal radiation. We assume that the sheet is permeable, allowing for the presence of either a suction or injection velocity. The purpose of this study is to gain insights into heat transfer and fluid dynamics, with different practical applications in engineering processes. The methodology includes mathematically modeling with partial differential equations, utilizing numerical methods for solution, and integrating nanofluid properties and boundary conditions. Based on the previously mentioned assumptions, we formulated a mathematical model in a differential form by employing boundary layer approximations. We have transformed the differential model into a dimensionless system by applying appropriate conversions. We utilized the numerical shooting technique within the Mathematica software package to solve the system of dimensionless differential equations. We have displayed the impacts of the key physical parameters that govern the mathematical model in both tabular and graphical formats. Investigating the flow of tangent hyperbolic nanofluids holds great importance as it can offer valuable insights for practical applications across engineering, nanotechnology, and thermal sciences. Notable findings from the study reveal that the velocity function showed decreased values due to higher values of the suction parameter, power law index parameter, and slip velocity parameter. The results of this research have been evaluated in the context of the previously established body of knowledge, demonstrating a significant agreement that supports the validity of the present solutions.

The one-dimensional Gross-Pitaevskii equation, under non-vanishing boundary condition, has a set of solitary solutions. The orbital stability of these solitons has been well established. However, the existing proof methods usually treat the cases of dark solitons and black solitons separately. Here we provide an alternative proof of this orbital stability result, which treats the two cases in a unified framework.

In this paper, we present a new proof and attain some new local Hamilton type gradient estimates for positive solutions to the parabolic equation ut=ΔVup+aulogu+buon a complete noncompact Riemannian manifold with k-Bakry-Émery Ricci curvature bounded from below, where p, a and b are some given constants. As applications, related local Hamilton type gradient estimates, some parabolic type Liouville theorems and Harnack inequalities for porous media type equation and fast diffusion type equation are established. Some known results were generalized by our results.

Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a left-Alia bialgebra that we call triangular. The notions of relative Rota-Baxter operators of left-Alia algebras and pre-left-Alia algebras are introduced to provide antisymmetric solutions of the left-Alia Yang-Baxter equation.

This paper examines the Cauchy problem described by the following equation: ∂tλ+1ϕ-Δϕ=∫0t(t-s)-γϕ(s,.)pds,ϕ(0,x)=ϕ0(x),ϕt(0,x)=ϕ1(x).(1)The equation involves the Caputo fractional derivative in time, denoted as ∂tλ+1ϕ. Additionally, The nonlinear term is determined by the memory term ∫0t(t-s)-γϕ(s,.)pds, where γ∈(0,1). Using the fixed point theorem, we establish the global existence of solutions to the Cauchy problem (1) for small initial data. We also investigate the impact of the nonlinearity parameter on the range of the exponent p and the estimation of the solutions.