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We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad–Shafranov equations and is inspired by the recent result (Gavrilov in A steady Euler flow with compact support. Geom Funct Anal 29(1):90–197, [Gav19]). [ABSTRACT FROM AUTHOR]

Nonlinear dynamical problems, characterized by unpredictable and chaotic changes among variables over time, pose unique challenges in understanding. This paper explores the coupled nonlinear (2+1)-dimensional complex modified Korteweg-de-Vries (cmKdV) equation-a fundamental equation in applied magnetism and nanophysics. The study focuses on dynamic behaviors, specifically examining bifurcations and equilibrium points leading to chaotic phenomena by introducing an external term to the system. Employing chaos theory, we showcase the chaotic tendencies of the perturbed dynamical system. Additionally, a sensitivity analysis using the Runge-Kutta method reveals the solution's stability under slight variations in initial conditions. Innovatively, the paper utilizes the planar dynamical system technique to construct various solitons within the governing model. This research provides novel insights into the behavior of the (2+1)-dimensional cmKdV equation and its applications in applied magnetism and nanophysics. [ABSTRACT FROM AUTHOR]

The goal of this paper is to find exact solutions to the improved modified Korteweg-de Vries (mKdV) equation with a conformable derivative using the Jacobi elliptic function expansion method. The improved mKdV equation is a prominent mathematical model in the realm of nonlinear partial differential equations, with widespread applicability in diverse scientific and engineering domains. This study leverages the well-known effectiveness of the Jacobi elliptic function expansion method in solving nonlinear differential equations, specifically focusing on the intricacies of the improved mKdV problem. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of these solutions, emphasizing their distinct features and depictions using Jacobi elliptic functions. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations. This work also contributes to our knowledge of the improved mKdV equation and shows that the Jacobi elliptic function expansion method is a useful tool for solving complex nonlinear situations. The study is enhanced with graphical illustrations of various solutions, which further enhance its analytical complexity. [ABSTRACT FROM AUTHOR]

The fundamental equations and some special solutions of contact mechanics for solid fractal surfaces are proposed in this paper. Firstly, as for fractal solid surface, the probability distribution equation of height without pressure, the probability distribution equation of vertical displacement with pressure and the probability distribution equation of contact stress are established. Secondly, considering the adhesion force, a special solution is given for the probability distribution equation of contact stress when the fractal index is 2. Finally, when the area ratio of multi-scale contact is 0.5, the influence of different derivative orders on the results is discussed. In this paper, a complete analytical framework is given for the static multi-scale contact mechanics. [ABSTRACT FROM AUTHOR]

Inspired by a pioneer work of Andersson and Kapitanski (Arch Ration Mech Anal 247(2):Paper No. 21, 76 pp, 2023), we prove the local well-posedness of the Cauchy problem of incompressible neo-Hookean equations if the initial deformation and velocity belong to H n + 2 2 + (R n) × H n 2 + (R n) ( n = 2 , 3 ), where n + 2 2 and n 2 is respectively a scaling-invariant exponent for deformation and velocity in Sobolev spaces. Our new observation relies on two folds: a reduction to a second-order wave-elliptic system of deformation and velocity; and a "wave-map type" null form intrinsic in this coupled system. In particular, the wave nature with "wave-map type" null form allows us to prove a bilinear estimate of Klainerman–Machedon type for nonlinear terms. So we can lower 1 2 -order regularity in 3D and 3 4 -order regularity in 2D for well-posedness compared with Andersson and Kapitanski (Arch Ration Mech Anal 247(2):Paper No. 21, 76 pp, 2023). [ABSTRACT FROM AUTHOR]

In this paper, we study fractional p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland's variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in R N ∖ Ω ‾ for fractional order p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

The (3 + 1) -dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients, which have very crucial applications in various areas, such as fluid dynamics, plasma physics and so on, are studied via unified and generalised unified method in this work. Solitary, soliton, elliptic, singular (periodic type) and non-singular (soliton-type) solutions of these two equations are extracted using the unified method. Otherwise, polynomial solutions in double-wave form and rational solutions in double-soliton form of the modified KdV equation with variable coefficients are acquired by exploiting the generalised unified method. The dynamical demeanours of these solutions help to comprehend the physical phenomena reflected by the equations, are depicted and analysed graphically for specific values of randomly undetermined parameters, which are diverse in each solution. The outcomes show that these two methods are quite trustworthy and effective to explore numerous solutions of nonlinear partial differential equations. We recognise that two approaches have never been utilised to study these two equations and work carried out in this paper is fresh and handy. Compared to previous methods, more comprehensive solutions can be obtained using them in this paper. [ABSTRACT FROM AUTHOR]

In 1973 Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with analytic nonlinearities are themselves analytic. On the other hand, the two authors of this paper more recently showed that bounded solutions of certain delay-differential equations, again with analytic nonlinearities, can be C ∞ smooth, yet not be analytic for certain ranges of the independent variable t. In this paper we extend the 1973 results to obtain analytic solutions of a broader class of delay-differential equations, including a wide variety of nonautonomous equations. Nevertheless, there are still equations with analytic nonlinearities possessing global bounded C ∞ solutions for which analyticity is unknown. This is the case, for example, for the equation y ′ (t) = g (y (t - 1)) + ε sin (t 2) where g is analytic and where y = 0 is a hyperbolic equilibrium when ε = 0 . [ABSTRACT FROM AUTHOR]

This paper presents a new approach to approximate the solution of Kepler's equation. It is found that by means of a series approximation, an angle identity, the application of Sturm's theorem, and an iterative correction method, the need to evaluate transcendental functions or query lookup tables is eliminated. The final procedure builds upon Mikkola's approach. Initially, a fifteenth-order polynomial is derived through a series approximation of Kepler's equation. Sturm's theorem is used to prove that only one real root exists for this polynomial for the given range of mean anomaly and eccentricity. An initial approximation for this root is found using a third-order polynomial. Then, a single generalized Newton–Raphson correction is applied to obtain fourteenth-place accuracies in the elliptical case, which is near machine precision. This paper will focus on demonstrating the procedure for the elliptical case, though an application to hyperbolic orbits through a similar methodology may be similarly developed. [ABSTRACT FROM AUTHOR]

In this paper, an exact elasticity solution of functionally graded material (FGM) rectangular plate under cylindrical bending is presented. The formulation is based on the displacement approach in which 3D elasticity equations are reduced to 2-D equations using plane-strain conditions. In the present formulation, an exact elasticity solution is feasible due to the consideration of simply supported boundary conditions with applied loads expressed in harmonic forms. Functionally graded material (FGM) infinite rectangular plate subjected to realistic transverse normal loads are analyzed. Solutions have been presented for FGM plate with various aspect ratios and gradation factors. Variation of displacements and stresses through the thickness has been investigated for the FGM plate subjected to single sinusoidal load, uniformly distributed load, hydrostatic load, and strip load. Validation of results is presented using reference results in the literature. The extensive results presented in this paper can be served as benchmark solutions for the assessment of improved plate theories. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a scenario concerning a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. By establishing suitable conditions, we have proved that a finite-time blow-up occurs in the solution. [ABSTRACT FROM AUTHOR]

Despite the historical position of the F-expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective–diffusive Cahn–Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe–Cr–Mo) and (Fe–X–Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty-two distinct types. Six of these are associated with Weierstrass-elliptic function solutions, while the remaining solutions are expressed in Jacobi-elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term 'new,' the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton-like and trigonometric-function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the Navier–Stokes–Korteweg equations governed by the evolution of compressible fluids with capillarity effects. We first investigate the global well-posedness of solution in the critical Besov space for large initial data. Contrary to pure parabolic methods in Charve et al. (Indiana Univ Math J 70:1903–1944, 2021), we also take the strong dispersion due to large capillarity coefficient κ into considerations. By establishing a dissipative–dispersive estimate, we are able to obtain uniform estimates and incompressible limits in terms of κ simultaneously. Secondly, we establish the large time behaviors of the solution. We would make full use of both parabolic mechanics and dispersive structure which implicates our decay results without limitations for upper bound of derivatives while requiring no smallness for initial assumption. [ABSTRACT FROM AUTHOR]

We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-killing condition at the interface. Both the coefficient describing the probability of killing and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in Komorowski et al. (Ann Prob 48:2290–2322, 2020), where the case of a non-degenerate probability of killing has been considered. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of backward problems for nonlinear fractional super-diffusion equations in Banach spaces. We consider the time fractional derivative in the sense of Caputo type. First, we establish some results for the existence of the mild solutions. Moreover, we obtain regularity results of the first order and fractional derivatives of mild solutions. These conclusions are mainly based on fixed point theorems and properties of α -resolvent family as well as Mittag-Leffler functions. Finally, two applications are provided to illustrate the efficiency of our results. [ABSTRACT FROM AUTHOR]

In this paper, we apply the steepest descent method to the Schläfli-type integral representation of the three-parameter Mittag-Leffler function (well-known as the Prabhakar function). We find the asymptotic expansions of this function for its large parameters with respect to the real and complex saddle points. For each parameter, we separately establish a relation between the variable and parameter of function to determine the leading asymptotic term. We also introduce differentiations of the three-parameter Mittag-Leffler functions with respect to parameters and modify the associated asymptotic expansions for their large parameters. As an application, we derive the leading asymptotic term of fundamental solution of the time-fractional sub-diffusion equation including the Bessel operator with large order. [ABSTRACT FROM AUTHOR]

In this paper, the exact explicit traveling wave solutions to the simplified modified Camassa–Holm (SMCH) equation. The SMCH equation is a significant nonlinear evolution equation because it can be used to describe certain physical processes in oceanography and fluid dynamics. The SMCH equation has important applications in mathematics, physics, and engineering. Using the modified auxiliary equation method and the extended G ′ G 2 -expansion method, we achieve precise solutions with traveling wave behavior. The proposed methods are applied for the first time in this work to examine the considered mathematical model. The solutions are extracted in terms of trigonometric, hyperbolic, and rational functions. The obtained results not only confirm the previously reported solutions of SMCH equation but also offer some new results. To clearly illustrate the physical implications of the examined equation, we provide graphical representations using 3D, 2D and contour plots for certain values of the parameters. The illustrations help to clarify the diverse characteristics and behaviors, associated with the solutions, providing useful insights for researchers as well as practitioners across a variety of scientific and technical disciplines. [ABSTRACT FROM AUTHOR]

This paper examines the complex Ginzburg Landau equation, which describes pulse propagation inside a fiber with the triple power law of self-phase modulation. Since the effect of parameter selection has become very important in relevant model studies recently, self-phase modulation has been added to the complex Ginzburg Landau equation, which has been studied in the literature, and it is aimed at investigating the analytical solutions of the presented equation. Adding the triple power law of the self-phase modulation parameter to the model, in addition to existing studies in the literature, emphasizes the innovative aspect and importance of the study. The first aim is to reveal bright and singular solitons using the new Kudryashov method. The new Kudryashov method is a technique that is frequently used in the literature, is effective for generating analytical solutions, provides ease of operation, and can be applied to a wide class of nonlinear partial differential equations. The second goal is to show that the obtained solutions have modulation stability. By using modulation instability analysis, the gain spectrum is formed for different parameter values. Graphic presentations support the findings. Moreover, bright and singular soliton portraits are demonstrated with 3D and 2D graphs. The novelty of the study lies in the fact that the relevant model has not been studied before with an effective method such as the new Kudryashov method, and the modulation instability has been studied for the first time. [ABSTRACT FROM AUTHOR]

The main aim of the presented paper is to obtain the exact solutions of the generalized Sasa–Satsuma equation using two different methods, the modified auxiliary equation method and (G ′ G , 1 G) -expansion method. As a result of these methods, new soliton solutions of the given model are extracted. These solutions include rational solutions, hyperbolic solutions and periodic solutions. The necessary constraint conditions for the existence and validity of the constructed solutions are also been provided. Moreover, for appropriate parametric values, acquired findings are displayed via contour, 2D and 3D graphics that show the physical relevance and dynamical behaviors of the proposed equation. The dynamics of constructed solutions of the proposed methods help a lot in describing the propagation of the femtosecond pulses in the systems of optical fiber transmission. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and concentration of ground state solutions for the following class of elliptic Kirchhoff–Boussinesq type problems given by Δ 2 u ± Δ p u + (1 + λ V (x)) u = f (u) in R 4 , where 2 < p < 4 , f ∈ C (R , R) is a nonlinearity which has subcritical or critical exponential growth at infinity and V ∈ C (R 4 , R) is a potential that vanishes on a bounded domain Ω ⊂ R 4. Using variational methods, we show the existence of ground state solutions, which concentrates on a ground state solution of a Kirchhoff–Boussinesq type equation in Ω. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the salp swarm optimization algorithm (SSOA)-based novel model reduction algorithm for simplifying commensurate and incommensurate fractional-order systems. In the case of commensurate fractional-order systems (CFOSs), the presented method converts it into a non-fractional-order system first. Then, we implement the SSOA with the stability equations to find a non-fractional-order approximant. Finally, the non-fractional-order approximant is transformed to determine the proposed reduced-order CFOS. It is also demonstrated using numerical examples that some existing methods fail to achieve the stability claim. On the other hand, the stability of the proposed fractional-order approximants is ascertained by incorporating stability equations along with SSOA. Further, the proposed method is extended for the model reduction in incommensurate and unstable fractional-order systems. In this case, the Oustaloup approximation is used along with the SSOA and stability equations to obtain the proposed non-fractional-order approximation. The simulation results corroborate the performance of the suggested method in both cases and establish its transcendency. The obtained results showcase the efficacy of optimization in simplification of fractional-order systems. The work also highlights the importance of order reduction in practical engineering and scientific applications of fractional-order systems. [ABSTRACT FROM AUTHOR]

The mixed equation, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy–Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah–Floer conjecture. [ABSTRACT FROM AUTHOR]

This paper is concerned with the backward problem of recovering initial value for a homogeneous time-space fractional pseudo parabolic differential equation. Since the problem is ill-posed, a version of modified quasi boundary value method is used as the method of regularization for obtaining stable approximations. Error analysis and parameter choice strategies are done for both a priori and a posteriori cases. It is shown that, the order of the convergence rate can exceed 2 3. Hence, the obtained rate improves upon the previously known rates of order 2 3 for the a priori case, and 1 2 for the a posteriori case for the considered problem. [ABSTRACT FROM AUTHOR]

In this paper, we apply the fractional homotopy perturbation transform method (FHPTM) to deliver an effective semi-analytical technique for determining fractional-order Kuramoto–Sivashinsky equations. The project technique combines the Laplace transform with the Caputo–Fabrizio fractional derivative of order α where α ∈ (0 , 1 ] . Fractional-order Kuramoto–Sivashinsky equation is indeed important in the field of nonlinear physics and mathematics. It is a fractional partial differential equation that describes the behaviour of waves in certain dissipative media, such as flames and chemicals. The FHPTM is described to be fast and accurate. Illustrative examples are included to demonstrate the efficiency and reliability of the presented techniques. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the conservation of energy and magnetic helicity of weak solutions for the three-dimensional electron magnetohydrodynamic (EMHD) equations. Firstly, we establish sufficient conditions to guarantee the energy (magnetic helicity) balance of weak solutions for the EMHD equations based on the magnetic field, which can be viewed as an analogue of famous Lions' energy balance criterion of the Navier–Stokes equations for the EMHD equations. Secondly, in the spirit of recent works due to Berselli and Chiodaroli (Nonlinear Anal 192: 111704, 2020), as reported by Berselli (Three-Dimensional Navier–Stokes Equations for Turbulence. Academic Press, London, 2021), Berselli (Mathematics 11(4): 1–16, 2023), Berselli (J Differ Equ 368: 350–375, 2023), Berselli and Georgiadis (Nonlinear Differ Equ Appl 31(33): 1–14, 2024), we present energy (magnetic helicity) preservation criteria in terms of the current density in this system for both the whole space and the torus cases. [ABSTRACT FROM AUTHOR]

In this paper, we prove the multiplicity and concentration behavior of complex-valued solutions for the following Kirchhoff–Schrödinger equation with magnetic field - (a ε 2 + b ε [ u ] A / ε 2) Δ A / ε u + V (x) u = f (| u | 2) u , x ∈ R 2 , where ε > 0 is a small parameter, the nonlinearity f is involved in critical exponential growth in the sense of Trudinger–Moser inequality and both V : R 2 → R and A : R 2 → R 2 are continuous potential and magnetic potential, respectively. Imposing a local constraint of potential V(x) first introduced from del Pino and Felmer, we get the multiplicity of solutions by way of the relationship between the number of the solutions and the topology of the set with V attaining the minimum. Our strategy of main proof is based on the variational methods combined with the penalization technique, the Trudinger–Moser inequality and Ljusternik–Schnirelmann theory, and our result is still new even without magnetic effect. [ABSTRACT FROM AUTHOR]

This paper deals with the semilinear Rayleigh–Stokes equation with the fractional derivative in time of order α ∈ (0 , 1) , which can be used to model anomalous diffusion in viscoelastic fluids. An operator family related to this problem is defined, and its regularity properties are investigated. We firstly give the concept of the mild solutions in terms of the operator family and then obtain the existence of global mild solutions by means of fixed point technique. Moreover, the existence and regularity of classical solutions are given. [ABSTRACT FROM AUTHOR]

In this paper, we provide sharp criteria of global attraction for a class of non-autonomous reaction–diffusion equations with delay and Neumann conditions. Our methodology is based on a subtle combination of some dynamical system tools and the maximum principle for parabolic equations. It is worth mentioning that our results are achieved under very weak and verifiable conditions. We apply our results to a wide variety of classical models, including the non-autonomous variants of Nicholson's equation or the Mackey–Glass model. In some cases, our technique gives the optimal conditions for the global attraction. [ABSTRACT FROM AUTHOR]

In this paper, we study a type of p-Laplacian equation - Δ p u = λ u p - 2 u + u q - 2 u , x ∈ R N , with prescribed mass ∫ R N | u | p 1 p = c > 0 , where 1 < p < q < p ∗ : = pN N - p , p < N , λ ∈ R is a Lagrange multiplier. Firstly, we prove the existence of normalized solutions to p-Laplacian equations and provide accurate descriptions; secondly, we discuss the existence of ground states; finally, we study the radial symmetry of normalized solutions in the mass supercritical case. Besides, we also study normalized solutions to p-Laplacian equation with a potential function V(x) - Δ p u + V (x) u p - 2 u = λ u p - 2 u + u q - 2 u , x ∈ R N , under different assumptions on q and the constraint norm c, we prove the existence, nonexistence, concentration phenomenon and exponential decay of normalized solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlinear stability of 2D Boussinesq-MHD equations around the hydrostatic equilibrium state (U ¯ , B ¯ , Θ ¯ , P ¯) = (0 , e 1 , y , 1 2 y 2) in two kinds of periodic domains: T × (0 , 1) and T × R . Due to the absence of dissipation of magnetic field, it brings great difficulties to obtain the uniformly a priori estimate of ∂ 1 b . The main technique is utilizing the couple of equations between u and b to obtain the anisotropic estimates of ( u , b , θ ) and then we obtain the expected stability results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of normalized solutions for the nonautonomous Schrödinger–Poisson equations - Δ u + λ u + | x | - 1 ∗ | u | 2 u = A (x) | u | p - 2 u , in R 3 , where λ ∈ R , A ∈ L ∞ (R 3) satisfies some mild conditions. Due to the nonconstant potential A, we use Pohozaev manifold to recover the compactness for a minimizing sequence. For p ∈ (2 , 3) , p ∈ (3 , 10 3) and p ∈ (10 3 , 6) , we adopt different analytical techniques to overcome the difficulties due to the presence of three terms in the corresponding energy functional which scale differently. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop and analyze an adaptive collocation method for Fredholm integral equations of the second kind, even if the equation exhibits localized rapid variations, steep gradients, or steep front. The strategy of the adaptive procedure is to transform the given equation into an equivalent one with a sufficiently smooth behavior in order to ensure the convergence of the Legendre spectral collocation method without dividing the domain of the integral equation, as usual, into the sub intervals. Existence and uniqueness of solutions are discussed. Convergence rates and error estimates are given for the numerical scheme in both L ∞ -norm and L 2 -norm. Finally, several numerical examples are provided to show that the proposed method is preferable to its classical predecessor and some other existing approaches. [ABSTRACT FROM AUTHOR]

In this paper, we develop a convex splitting algorithm for the time-dependent Ginzburg-Landau equation, which can preserve both the energy stability and maximum bound principle. The basic idea of the convex splitting method is to decompose the energy functional into the convex part and the concave part. The term corresponding to the convex part of the equation is implicitly treated, and the concave part is explicitly processed. The backward Euler time discretizing method is chosen for the time-dependent Ginzburg-Landau equation. The theoretical analysis proves that the convex splitting method can preserve the maximum bound principle and energy stability. The numerical results show that the numerical algorithm is stable. [ABSTRACT FROM AUTHOR]

In this paper, we focus on solving the generalized absolute value equations (GAVE). We present a new method named as modulus-based block triangular splitting iteration (MBTS) method based on the block matrix structure resulting from the transformation of the GAVE into two equations. This method is developed by decomposing the matrix into diagonal and triangular matrices, as well as applying a series of suitable combination and modification techniques. The advantage of the MBTS method is that it is not necessary to solve the inverse of the coefficient matrix of the linear equation system during each iteration, which greatly improves the computational speed and reduces its storage requirements. In addition, we present some convergent theorems proving by different techniques and the estimate of the required number of iteration steps. Furthermore, in the accompanying corollaries, we provide some estimations for choosing appropriate parameter values. Finally, we validated the effectiveness and efficiency of our newly developed method through two numerical examples of the GAVE. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the existence of solutions to a double-phase fractional equation of the Kirchhoff type in Musielak-Orlicz Sobolev space with variable exponents. Our approach is mainly based on the sub-supersolution method and the mountain pass theorem. [ABSTRACT FROM AUTHOR]

The doubly dispersive (DD) equation finds extensive utility across scientific and engineering domains. It stands as a significant nonlinear physical model elucidating nonlinear wave propagation within the elastic inhomogeneous Murnaghan's rod (EIMR). With this in mind, we have focused on the integration of the DD model and the modified Khater (MK) method. Through the wave transformation, this model is effectively converted into an ordinary differential equation. In this paper, the goal of our work is to explore new wave solutions to the DD model by using the MK scheme. These solutions provide extremely helpful insights into the operation of the system. The three-dimensional (3D) plot and two-dimensional (2D) combined plot via the impacts of the parameters are provided for various parameters in this manuscript. We also discussed the dynamical properties of the model, which are accomplished through the bifurcation analysis, and also found the Hamiltonian function. This research makes a substantial contribution to the area by increasing our understanding of wave solutions in the DD, introducing novel investigation tools, and carrying out an in-depth investigation of the bifurcation and stability aspects of the system. As a direct result of this research, novel openings have been uncovered for further investigation and application in the various disciplines of science and engineering. [ABSTRACT FROM AUTHOR]

The authors considered the problem of optimal control of loading points and the corresponding reaction functions described by a loaded parabolic equation. They obtained optimality conditions for control actions. The authors used the objective functional gradient formulas contained in these conditions in the algorithm for numerically solving the problem of controlling the movement of loading points and corresponding reaction functions based on first-order optimization methods. The paper provides the results of numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, the (2 + 1)-dimensional variable coefficients equation which describes the thermophoric wave motion of wrinkles in graphene sheets (2D-vGS) is studied, where it has many applications in 2D optics, nanophotonic, and nanoelectronics. A direct simplified Hirota's bilinear method is generalized to find the bilinear form of the 2D-vGS equation. Accordingly, one, two, and three soliton wave solutions indicate that our studied equation is fully integrable and has n-soliton solutions. Moreover, we have focused on the study of two and three solitons interactions, this leads to the identification of two distinct solution types, the Y-shape soliton and fork- shape soliton, which can be clearly distinguished from the 3D plots and density plots. These solutions are characterized by a rich spectrum of collision dynamics and encompassing phenomena such as fusion and fission. The nonlinear properties of the two and three soliton solutions could be useful for farther applications in 2D optics like metamaterials with exotic optical properties and ultra-compact and efficient photonic devices. [ABSTRACT FROM AUTHOR]

This research paper's primary goal is to find fresh approaches to the Hirota–Maccari system. This system explains the dynamical features of the femto-second soliton pulse in single-mode fibers. The bright soliton, dark soliton, dark-bright soliton, dark singular, bright singular, periodic soliton, and singular solutions are developed utilizing the modified auxiliary equation technique. To make the physical significance of each unique solution clearer, it is mapped in both 2D and 3D. The primary Hirota–Maccari system is being verified by all new solutions, and the constraint condition is also provided. The obtained optical solitons may be important for the analysis of nonlinear processes in optic fiber communication and signal processing. [ABSTRACT FROM AUTHOR]

This paper deals with the extraction of exact solutions of nonlinear extended quantum Zakharov–Kuznetsov equation by implementing well founded approaches like generalized exponential rational function method and G ′ G , 1 G expansion method. Singular, periodic, hyperbolic and rational function solutions are achieved for particular values of parameters. The obtained solutions are discussed graphically by plotting 2D line plots and 3D surface plots. This research findings demonstrate that applying these approaches is a feasible means of solving numerous precise PDE solutions. The computations are worked out in this script with the aid of Mathematica software. [ABSTRACT FROM AUTHOR]

The analytical study of Boiti-Leon-Manna-Pempinelli (BLMP) equation is presented in this research paper. In this study, two exact methods are utilized to attain the exact solution of proposed equation. The generalized projective Riccati equations method and modified auxiliary equation method are simple and effective techniques, which have been used to attain the exact soliton solutions of BLMP equation. Some novel exact solution of BLMP equation are acquired using of proposed methods. The obtained solutions contain rational, geometric, hyperbolic functions. The graphical simulations of attained solutions are represented by plotted graphs. The plotted graphs show different solitons patterns such as kink solitons, anti-kink soliton, dark singular soliton, bright singular soliton, dark-bright singular solition and some other singular solitons. Mathematical modeling, analysis of physical phenomena and dynamical processes can yield solutions that enhance our understanding of their dynamics, which can be leveraged to gain valuable insights into the behavior and characteristics of these systems. [ABSTRACT FROM AUTHOR]

In this paper, we analyze Riemann–Liouville (R-L) time-fractional (2 + 1) dimensional extended quantum Zakharov–Kuznetsov (EQZK) equation by using the Lie symmetry method which arises in hydrodynamic that describes the nonlinear propagation of the quantum ion-acoustic waves. By using its symmetry, we convert the equation under consideration to a fractional order non-linear ordinary differential equation (ODE). In this reduced ODE, we use a special type of derivative which is known as Erdélyi–Kober (EK) derivative. This enables us to obtain explicit solutions with convergence analysis of the considered problem. By using Ibragimov's conservation laws theorem, we compute the conservation laws of the problem under investigation. Moreover, by employing the two potent methods explicit power series and ( 1 G ′ )-expansion technique, we get the explicit solutions to the problem under discussion. This analysis leads to the derivation of various key findings, including the identification of symmetries, the establishment of similarity reductions involving the EK fractional differential operator, the determination of exact solutions, and the formulation of conservation laws for the considered equation. We have confidence that these remarkable findings can provide valuable insights and contribute to the exploration of additional evolutionary mechanisms associated with the studied equation. [ABSTRACT FROM AUTHOR]

In the present paper we propose a novel analytical technique to obtain the solution of nonlinear partial differential equations. Additionally, we also implement the proposed method to find out the solution of fractional Kudryashov–Sinelshchikov equation. Since the Kudryashov–Sinelshchikov equation exhorts the pressure waves in mixture of liquid gas bubble while considering the viscosity and heat transport. Here regularized version of Hilfer–Prabhakar derivative of fractional order is utilized to model the problem. The obtained results are presented graphically for some specific values of constants and for distinct values of fractional order at different stages of time. The graphical behaviour of results show that the proposed method is very efficient to solve the nonlinear partial differential equations and reliable. [ABSTRACT FROM AUTHOR]

This paper deals with a new variant of the Biswas–Milovic equation, referred to as the perturbed Biswas–Milovic equation with parabolic-law nonlinearity in spatio-temporal dispersion. To our best knowledge, the considered equation which models the pulse propagation in optical fiber is studied for the first time, and the abundant optical solitons are successfully obtained utilizing the auxiliary equation method. Utilizing a wave transformation technique on the considered Biswas–Milovic equation, and by distinguishing its real and imaginary components, we have been able to restructure the considered equation into a set of nonlinear ordinary differential equations. The solutions for these ordinary differential equations, suggested by the auxiliary equation method, include certain undetermined parameters. These solutions are then incorporated into the nonlinear ordinary differential equation, leading to the formation of an algebraic equation system by collecting like terms of the unknown function and setting their coefficients to zero. The undetermined parameters, and consequently the solutions to the Biswas–Milovic equation, are derived by resolving this system. 3D, 2D, and contour graphs of the solution functions are plotted and interpreted to understand the physical behavior of the model. Furthermore, we also investigate the impact of the parameters such as the spatio-temporal dispersion and the parabolic nonlinearity on the behavior of the soliton. The new model and findings may contribute to the understanding and characterization of the nonlinear behavior of pulse propagation in optical fibers, which is crucial for the development of optical communication systems. [ABSTRACT FROM AUTHOR]

This paper investigates a novel method for exploring soliton behavior in ion transport across biological membranes. This study uses the Hirota bilinear transformation technique together with the Poisson–Nernst–Planck equation. A thorough grasp of ion transport dynamics is crucial in many different scientific fields since biological membranes are important in controlling the movement of ions within cells. By extending the standard equation, the suggested methodology offers a more thorough framework for examining ion transport processes. We examine a variety of ion-acoustic wave structures using the Hirota bilinear transformation technique. The different forms of solitons are obtained including breather waves, lump waves, mixed-type waves, periodic cross-kink waves, M-shaped rational waves, M-shaped rational wave solutions with one kink, and M-shaped rational waves with two kinks. It is evident from these numerous wave shapes that ion transport inside biological membranes is highly relevant, and they provide important insights that may have an impact on various scientific disciplines, medication development, and other areas. This extensive approach helps scholars dig deeper into the complexity of ion transport, illuminating the complicated mechanisms driving this essential biological function. Additionally, to show the physical interpretations of these solutions we construct the 3D and their corresponding contour plots by choosing the different values of constants. So, these solutions give us the better physical behaviors. [ABSTRACT FROM AUTHOR]

We consider the balanced pantograph equation (BPE) y ′ x + y x = ∑ k = 1 m p k y a k x , where ak, pk > 0 and ∑ k = 1 m p k = 1 . It is known that if K = ∑ k = 1 m p k ln a k ≤ 0 then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for K > 0 these solutions exist. In the present paper, we deal with a BPE of mixed type, i.e., a1< 1 < am, and prove that, in this case, the BPE has a nonconstant solution y and that y(x) ~ cxσ as x → ∞, where c > 0 and σ is the unique positive root of the characteristic equation P s = 1 - ∑ k = 1 m p k a k - s = 0 . We also show that y is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as x → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we study fully nonlinear equations of Krylov type in conformal geometry on compact smooth Riemannian manifolds with totally geodesic boundary. We prove the a priori estimates for solutions to these equations and establish an existence result. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the hypersurface that can be locally represented as a graph and satisfies a class of Hessian quotient type curvature equations. We establish interior curvature estimates under the condition of 0 ≤ l < k ≤ C n - 1 p - 1 . As an application, we prove Bernstein type theorem for this type curvature equation. We also focus on closed star shaped hypersurface satisfying this type curvature equation and obtain the global curvature estimation. [ABSTRACT FROM AUTHOR]

The present paper shows that local similarity theories, proposed for the strongly-stratified boundary layers, can be derived as invariant solutions defined under the Lie-group theory. A system truncated to the mean momentum and buoyancy equations is considered for this purpose. The study further suggests how similarity functions for the mean profiles are determined from the vertical fluxes, with a potential dependence on a measure of the anisotropy of the system. A time scale that is likely to characterize the transiency of a system is also identified as a non-dimensionalization factor. [ABSTRACT FROM AUTHOR]

This paper discusses about a new compact 2-level implicit numerical method in the form of exponential approximation for finding the approximate solution of nonlinear fourth order Kuramoto–Sivashinsky and Fisher–Kolmogorov equations, which have applications in chemical engineering. The described method has an accuracy of temporal order two and a spatial order three (or four) on a variable (or constant) mesh. The approach has been demonstrated to be applicable to both non-singular and singular issues. This article has established the stability of the current technique. The suggested approach is used to solve several benchmark nonlinear parabolic problems associated in chemistry and chemical engineering, and the computed results are compared with the existing results to demonstrate the proposed method's superiority. [ABSTRACT FROM AUTHOR]