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Denote by h = h (− p) the class number of the imaginary quadratic field ℚ (− p) with p prime. It is well known that h = 1 for p ∈ { 3 , 7 , 1 1 , 1 9 , 4 3 , 6 7 , 1 6 3 }. Recently, all the solutions of the Diophantine equation x 2 + p s = 4 y n with h = 1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation x 2 + p s = 2 r y n in unknown integers (x , y , s , r , n) , where s ≥ 0 , r ≥ 3 , n ≥ 3 , h ∈ { 1 , 2 , 3 } and gcd (x , y) = 1. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Boltzmann equation with external force in the whole space, where the collision kernel is assumed to be hard potential and cutoff. We prove that the solutions of such Boltzmann equations are L p (1 ≤ p < ∞) stable under the perturbation of external force. Our estimate is based on the gradient estimate of the solution. The key step of this paper is to estimate the solutions of the equations propagate in different forward bi-characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we explore how to generate solitary, peakon, periodic, cuspon and kink wave solution of the well-known partial differential equation Korteweg–de Vries (KdV) by using exp-function and modified exp-function methods. The presented methods construct more efficiently almost all types of soliton solution of KdV equation that can be rarely seen in the history. These methods appear to be straightforward and symbolic calculations are used to solve the problem. All resulting answers are verified for accuracy using the symbolic computation program with M a p l e. To show the physical appearance of the model, 3D plots of all the generated solutions are then displayed. The obtained solutions revealed the compatibility of the proposed techniques which provide the general solution with some free parameters. This is the key benefit of these methods over the other methods. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of O (h k) in the energy norm, where k represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach. [ABSTRACT FROM AUTHOR]

In this paper, under the observation of extended modified rational expansion method based on symbolic computation, the multiple solitary wave solutions for nonlinear two-dimensional Jaulent–Miodek Hierarchy (JMH) equation are constructed. In this investigation, we use the computer software Mathematica for the construction of multiple solitary wave solutions. The interested and important things in this work are the multiple solitary wave solutions which have various kinds of physical structures such as kink soliton, periodic traveling wave, bright soliton, anti-kink soliton, dark soliton, combined bright and dark solitons, topological soliton and peakon soliton. We are sure that the various kinds of soliton solutions are found first time by using one method in the existing literature works. On the basis of this research, we can say that the applied technique is very efficient, reliable, fruitful and powerful. The constructed soliton solutions for nonlinear JMH equation will play an important role in the investigation of different physical phenomena in nonlinear sciences. [ABSTRACT FROM AUTHOR]

In general, a perfect fluid spacetime is not a generalized Robertson–Walker spacetime and the converse is also not true. In this paper, it is shown that if a perfect fluid spacetime satisfies the critical point equation, then either the spacetime becomes a generalized Robertson–Walker spacetime and represents dark era or the vorticity of the fluid vanishes as well as the spacetime is expansion free. Besides, we prove that if a generalized Robertson–Walker spacetime with constant scalar curvature satisfies the critical point equation, then the spacetime becomes a perfect fluid spacetime. Next, the existence of critical point equation is established by a non-trivial example. Finally, we discuss the critical point equation in f (r) -gravity. For the model f (r) = r − α (1 − e − r α ) (α = constant and r is the scalar curvature of the spacetime), various energy conditions in terms of the scalar curvature are examined and state that the Universe is in an accelerating phase and satisfies the weak, null, dominant, and strong energy conditions. [ABSTRACT FROM AUTHOR]

In this paper, we first study the dual fractional parabolic equation ∂ t α u (x , t) + (− Δ) s u (x , t) = f (u (x , t)) in  B 1 (0) × ℝ , subjected to the vanishing exterior condition. We show that for each t ∈ ℝ , the positive bounded solution u (⋅ , t) must be radially symmetric and strictly decreasing about the origin in the unit ball in ℝ n . To overcome the challenges caused by the dual nonlocality of the operator ∂ t α + (− Δ) s , some novel techniques were introduced. Then we establish the Liouville theorem for the homogeneous equation in the whole space ∂ t α u (x , t) + (− Δ) s u (x , t) = 0 in  ℝ n × ℝ. We first prove a maximum principle in unbounded domains for antisymmetric functions to deduce that u (x , t) must be constant with respect to x. Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation ∂ t α u (t) = 0 in  ℝ. To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator ∂ t α , we bring in some new ideas and simpler approaches. Instead of disturbing the antisymmetric function, we employ a perturbation technique directly on the solution u (t) itself. This method provides a more concise and intuitive way to establish the Liouville theorem for one-sided operators ∂ t α , including even more general Marchaud time derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we seek to find solutions of the local time fractional Telegraph equation (LTFTE) by employing the local time fractional reduced differential transform method (LTFRDTM). This method produces a numerical approximate solution having the form of an infinite series that converges to a closed form solution in many cases. We apply LTFRDTM on four different LTFTEs to examine the efficiency of the proposed method. The yielded results established the effectiveness of LTFRDTM as a reliable and solid approach for obtaining solutions of LTFTEs. The solutions coincided with the exact solution in the ordinary case when μ = 1. It also required minimal amount of computational work and saved a lot of time. [ABSTRACT FROM AUTHOR]

In this paper, a reliable and effective local discontinuous Galerkin (LDG) scheme for numerically solving the classical Rosenau–Hyman equation with non-periodic boundary conditions has been proposed. This study employs the third-order nonlinearly stable total variation diminishing Runge–Kutta method and the LDG method, respectively, to discretize the temporal and spatial derivatives. Finally, numerical simulations are performed on various test problems and compared with the exact results as well as results produced by a few other numerical methods, to analyze the reliability and efficiency of the proposed method. The results generated, which validate the expected order of accuracy, are presented through multiple tables. In addition, several graphical representations of the problem are presented to depict the behavior of the solution. [ABSTRACT FROM AUTHOR]

The aim of this review paper is to revisit the geometric framework of the Teukolsky equation in a form that is suitable for analysts working on this equation. We introduce spinor bundles, the Newman–Penrose formalism and the Geroch–Held–Penrose (GHP) formalism. In particular, we develop the case of Kerr spacetimes, for which we provide detailed computations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the monotonicity and one-dimensional symmetry of solutions for parabolic equations related to the weighted fractional Laplacian on the whole space. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, and then carry out the sliding method to obtain monotonicity and one-dimensional symmetry of solutions. [ABSTRACT FROM AUTHOR]

This paper addresses the new (3 + 1) -dimensional Mikhailov–Novikov–Wang (MNW) equation with arbitrary order derivative and presents novel exact solutions of it by implementing exp (− φ (ξ)) -expansion, modified Kudryashov, generalized (G ′ / G) -expansion, and modified extended tanh-function methods. This equation emphasizes significant connection between the integrability and water waves' phenomena. Employing the conformable derivative definition, a variety of soliton (bright, dark, anti-kink) solutions of the model are obtained. Therefore, it would appear that these approaches might yield noteworthy results in producing the exact solutions to the fractional differential equations in a wide range. In addition, 2D, 3D, and contour plots of the solutions are drawn for specific values to demonstrate the physical behaviors of the solutions. [ABSTRACT FROM AUTHOR]

Let (M n , F , m) be a compact Finsler manifold. This paper will obtain a global estimate and a differential Harnack inequality for positive solutions to the Newell–Whitehead–Segal equation ∂ t u (x , t) = Δ m u (x , t) + a u − b u 3 ,   (x , t) ∈ M × [ 0 , T ] , on compact Finsler manifolds, where a , b > 0 are real constants. [ABSTRACT FROM AUTHOR]

This paper discusses the wave propagation to the nonlinear Sharma–Tasso–Olver–Burgers (STOB) equation which is used as the governing model in different fields. Natural phenomena are typically complex and nonlinear, defying simple linear superposition. Researchers have been studying a wide range of natural phenomena in depth, and nonlinear science has gradually become a part of people's consciousness. One of the most significant research questions in nonlinear science centers around the nonlinear evolution equation and its precise solution. We have secured different shapes of the solitary wave solutions including kink-type, shock-type and combined solitary wave solutions with the assistance of recently developed integration tool, namely the new extended direct algebraic method (NEDAM). Additionally, the solutions for the hyperbolic, exponential and trigonometric functions are retrieved. Moreover, based on a comparison of our results to those that are well known, the study indicates that our solutions are innovative. Using proper parameters in numerical simulations and physical explanations, it is possible to demonstrate the significance of the results. The results of this research can improve the nonlinear dynamic behavior of a system and indicate that the methodology employed is adequate. It is proposed that the offered method can be utilized to support nonlinear dynamical models applicable to a wide variety of physical situations. We hope that a wide spectrum of engineering model professionals will find this study to be beneficial. [ABSTRACT FROM AUTHOR]

The Novikov equation is a generalization of the Camassa–Holm one. It models the propagation of unidirectional shallow water waves on a flat bottom. In this paper, we prove the well-posedness of H 2 and H 3 solutions of the Cauchy problem, associated with this equation. [ABSTRACT FROM AUTHOR]

In this paper, we continue the rigorous study of classical effective field theories (EFTs) that was recently initiated in the work of Reall and Warnick. We study a system with one light and one heavy field with cubic coupling and prove global existence [of the ultraviolet (UV) solution] under an effective norm in the high mass limit. Furthermore, we prove that the global solution linearly scatters (in Lax–Philips sense) and that this final state has an expansion in inverse powers of the mass, and can be recovered from the EFT equation alone. [ABSTRACT FROM AUTHOR]

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L (p , q) , KBSM(L (p , q)), for q ≠ 0. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, TL 1 , n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 , n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM(L (p , q)), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L (p , q) but not in ST, and which reflect the surgery description of L (p , q) , obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM(L (p , q)). We first present the solution for the case q = 1 , which corresponds to obtaining a new basis, ℬ p , for KBSM(L (p , 1)) with (⌊ p / 2 ⌋ + 1) elements. We note that the basis ℬ p is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case q > 1 , we first show how the new basis ℬ p of KBSM(L (p , 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of multiple non-radial positive and sign-changing solutions for the following Kirchhoff equation: − a + b ∫ ℝ 3 | ∇ u | 2 Δ u + (1 + λ Q (x)) u = | u | p − 2 u , in ℝ 3 , (0. 1) where a , b > 0 are constants, p ∈ (2 , 6) , λ is a parameter, and Q (x) is a potential function. Under the assumption on Q (x) with exponential decay at infinity, we construct multi-peak positive and sign-changing solutions for problem (0.1) as λ → ∞ (or 0), where the peaks concentrate at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we conduct a comprehensive exploration of the relativistic quantum dynamics of spin-0 scalar particles, as described by the Duffin–Kemmer–Petiau (DKP) equation, within the framework of a magnetic space-time. Our focus is on the Bonnor–Melvin–Lambda (BML) solution, a four-dimensional magnetic universe characterized by a magnetic field that varies with axial distance. To initiate this investigation, we derive the radial equation using a suitable wave function ansatz and subsequently employ special functions to solve it. Furthermore, we extend our analysis to include Duffin–Kemmer–Petiau oscillator fields within the same BML space-time background. We derive the corresponding radial equation and solve it using special functions. Significantly, our results show that the geometry's topology and the cosmological constant (both are related to the magnetic field strength) influence the eigenvalue solution of spin-0 DKP fields and DKP-oscillator fields, leading to substantial modifications in the overall outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we use new extended generalized Kudryashov and improved tan (ϑ 2) expansion approaches to investigate Kerr law nonlinearity in the extended Gerdjikov–Ivanov equation in a dense wavelength division multiplexed system. These methods rely on a traveling wave transformation and an auxiliary equation. These approaches successfully extract trigonometric, rational and hyperbolic solutions, along with some appropriate conditions imposed on parameters. To explain the dynamics of soliton profiles, a graphical description of newly discovered solutions is also presented, which exhibits distinct physical significance. The considered methods are recognized as useful and influential tools for creating solitary wave solutions to nonlinear problems in the mathematical sciences. [ABSTRACT FROM AUTHOR]

In this paper, we study the decaying dynamics in the mirror-field interaction by means of the intrinsic decoherence scheme. Factorization of the mirror-field Hamiltonian with the use of displacement operators allows us to calculate the explicit solution to Milburn's equation for arbitrary initial conditions. We show expectation values, correlations, and Husimi functions for the solutions obtained. [ABSTRACT FROM AUTHOR]

In this paper, the Titchmarsh–Weyl theory of the impulsive q -Dirac equation is studied. [ABSTRACT FROM AUTHOR]

Algebraic operations on generalized hesitant fuzzy numbers are key tools to address the problems with decision uncertainty. In this paper, by studying the arithmetic operations on generalized trapezoidal hesitant fuzzy numbers, modified arithmetic operations are introduced for this class of numbers so that, using these arithmetic operations, the multiplication and division of two generalized trapezoidal hesitant fuzzy numbers are always generalized trapezoidal hesitant fuzzy numbers. Furthermore, a generalized trapezoidal hesitant fuzzy number raised to the power of a real number is a generalized trapezoidal hesitant fuzzy number, and in the defined division, the case where the denominator becomes zero is not considered. Numerical examples are used to show the shortcomings of the previous arithmetic operations as well as the efficiencies of the arithmetic operations proposed in this research for generalized trapezoidal hesitant fuzzy numbers. Finally, the application of the proposed new arithmetic operations to generalized trapezoidal hesitant fuzzy numbers in solving the generalized trapezoidal hesitant fully fuzzy equation is discussed. [ABSTRACT FROM AUTHOR]

The Fokker–Planck equation (FPE) is an important deterministic tool for investigating stochastic dynamical systems. In this paper, we consider the space-time fractional FPE driven by multiplicative Marcus Lévy noises. Efficient numerical schemes are presented to solve the equations. Stability and convergence of the methods are also discussed. We give some numerical experiments to validate our schemes, and examine the effects of parameters on solutions. Additionally, we analyze the maximal likely trajectories and the critical time for the change of the most probability location. [ABSTRACT FROM AUTHOR]

On pp. 243 and 244 of his second notebook, Ramanujan recorded seven modular equations of degree 11 followed by two more which he had crossed out. The first modular equation in the list was proved by Berndt using theta function identities. The same was also proved by Venkatachaliengar in a different way. To the best of our knowledge, the only available proofs to the other six modular equations in the list are by Berndt. These proofs employ the theory of modular forms. Interestingly these are the first set of modular equations, the proofs to which by Berndt take a shift from the elementary algebraic methods to the theory of modular forms. In this paper, we reprove all these modular equations of degree 11 (except the first one in the list) by elementary algebraic techniques. First, we use two series identities due to Ye and Liu to prove one of the modular equations and its reciprocal. The remaining modular equations are proved by the method of parametrization. [ABSTRACT FROM AUTHOR]

The fractional derivatives have been widely applied in many fields and has attracted widespread attention. This paper extracts a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKe) with the local fractional derivative (LFD) for the first time. Two special functions, namely, the LT δ (Ξ δ) and LC δ (Ξ δ) functions that are derived on the basis of the Mittag-Leffler function (MLF) defined on the Cantor set (CS), are employed to construct the auxiliary trial function to look into the exact solutions (ESs). Aided by Yang's non-differentiable (ND) transformation, six groups of the ND ESs are found. The ND ESs on the CS for δ = ln 2 / ln 3 are depicted graphically. Additionally, as a comparison, the ESs of the classic (3+1)-dimensional MZKe for δ = 1 are also illustrated. The outcomes reveal that the derived method is powerful and effective, and can be used to deal with the other local fractional PDEs. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a new class of nonlinear multi-term impulsive anti-periodic boundary value problems involving Caputo type fractional q -derivative operators of different orders and the Riemann–Liouville fractional q -integral operator. The uniqueness of solutions to the given problem is proved with the aid of Banach's fixed point theorem. Applying a Shaefer-like fixed point theorem, we also obtain an existence result for the problem at hand. Examples are constructed for illustrating the obtained results. The paper concludes with certain interesting observations concerning the reduction of the results proven in the paper to some new results under an appropriate choice of the parameters involved in the governing equation. [ABSTRACT FROM AUTHOR]

This paper studies the Lax pair (LP) of the (1 + 1) -dimensional Benjamin–Bona–Mahony equation (BBBE). Based on the LP, initial solution and Darboux transformation (DT), the analytic one-soliton solution will also be obtained for BBBE. This paper contains a bunch of soliton solutions together with bright, dark, periodic, kink, rational, Weierstrass elliptic and Jacobi elliptic solutions for governing model through the newly developed sub-ODE method. The BBBE has a wide range of applications in modeling long surface gravity waves of small amplitude. In addition, we will evaluate generalized breathers, Akhmediev breathers and standard rogue wave solutions for stated model via appropriate ansatz schemes. [ABSTRACT FROM AUTHOR]

This paper adopts the rational extended sine-cosine and cosh-sinh methods to construct the Bogoyavlenskii equation's exact solutions. To the best of our knowledge, the Bogoyavlenskii equation has not been investigated by aforementioned techniques. In this paper, we find the precise traveling wave solutions of the Bogoyavlenskii equation. Finally, 3D and 2D graphics of the obtained solutions are illustrated for the applicability and reliability of the proposed strategy for various special values. [ABSTRACT FROM AUTHOR]

This paper derives a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation based on the conformable fractional derivative for the first time. Some new types of the fractal traveling wave solutions are successfully constructed by applying a novel approach which is called the fractal semi-inverse variational method. To our knowledge, the obtained results are all new and have not reported in the other literature. In addition, the dynamic characteristics of the different solutions on the fractal space are discussed and presented via the 3D plots, 2D contour and 2D curves. It can be found that: (1) The fractal order can not only affect the peak value of the fractal traveling waves, but also affect the wave structures, that is, the smaller the fractional order value is, the more curved the waveform is, and the slower waveform changes. (2) In the fractal space, the fractal wave keeps its shape unchanged in the process of the propagation and still meets the energy conservation. The methods in this paper can be used to study the other fractal PDEs in the physics, and the findings are expected to bring some new thinking and inspiration toward the fractal theory in physics. [ABSTRACT FROM AUTHOR]

The variational theory has triggered skyrocketing interest in the solitary theory, and the semi-inverse method has laid the foundation for the search for a variational formulation for a nonlinear system. This paper gives a brief review of the last development of the fractal soliton theory and discusses the variational principle for fractal Boussinesq-like B (m , n) equation in the literature. The paper establishes a variational formulation for B (m , 1) equation to show the effectiveness of the semi-inverse method, and a general trial-Lagrange function with two free parameters is established for B (m , n) equation, the identification of the unknown parameters and the unknown function involved in the trial-Lagrange function is shown step by step. This paper opens a new path for the fractal variational theory. [ABSTRACT FROM AUTHOR]

This paper is a key contribution with respect to the applications of solitary wave solutions to the unique solution in the presence of the auxiliary data. Hence, this study provides an insight for the unique selection of solitons for the physical problems. Additionally, the novel numerical scheme is developed to compare the result. Further, this paper deals with the stochastic Fisher-type equation numerically and analytically with a time noise process. The nonstandard finite difference scheme of stochastic Fisher-type equation is proposed. The stability analysis and consistency of this proposed scheme are constructed with the help of Von Neumann analysis and Itô integral. This model is applicable in the wave proliferation of a viral mutant in an infinitely long habitat. Additionally, for the sake of exact solutions, we used the Riccati equation mapping method. The solutions are constructed in the form of hyperbolic, trigonometric and rational forms with the help of Mathematica 11.1. Lastly, the graphical comparisons of numerical solutions with exact wave solution with the help of Neumann boundary conditions are constructed successfully in the form of 3D and line graphs by using different values of parameters. [ABSTRACT FROM AUTHOR]

In this paper, the complete discrimination system for the polynomial method is applied to solve the Wu–Zhang system, and all the possible exact solutions are obtained, these exact solutions can be applied to the exploration of nonlinear physical phenomena, the method in this paper is different from the existing literature studies on the Wu–Zhang equation. By taking different parameters, interesting graphs are plotted for all the obtained solutions. The results confirm that the proposed method is effective and can be used to solve a variety of nonlinear consistency time fractional partial differential equations. [ABSTRACT FROM AUTHOR]

Considering the importance of using nonlinear evolution equations in the investigation of many natural phenomena, in this paper, we consider the (2 + 1) -dimensional Date–Jimbo–Kashiwara–Miwa ((2 + 1) -dimensional DJKM) equation, we will investigate the solutions for this equation. Using the multiple exp functions method, we obtain analytical solutions for this equation, which are one-soliton, two-soliton and three-soliton solutions and these solutions include three categories of soliton wave solutions, i.e., one-wave solutions, two-wave solutions and three wave solutions. We have performed all calculations with a computer algebra system such as Maple and have also provided a graphical representation of the obtained solutions. [ABSTRACT FROM AUTHOR]

This paper tackles the recently introduced (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation (4D-BLMPE) utilized to model wave phenomena in incompressible fluid and fluid mechanics. Modified extended tanh expansion method (METEM) and the new Kudryashov scheme are implemented to produce analytical soliton solutions for the presented equation. The traveling wave transformation is constructed, and the homogeneous balance principle is utilized to apply the two proposed techniques. Furthermore, the flat-kink, smooth-kink, singular, and periodic singular solutions are successfully extracted. Some produced solutions are illustrated graphically to understand the physical meaning of the presented model. Moreover, for the first time in this study, the effect of model parameters on kink soliton dynamics is examined, and graphical representations are depicted and interpreted. [ABSTRACT FROM AUTHOR]

In this paper, the Titchmarsh–Weyl theory of the impulsive q -Dirac equation is studied. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the Zakharov–Kuznetsov (ZK) equation in the n -dimensional setting with n ≥ 2 and investigate its smoothness properties. We extend the well-known regularity propagation phenomenon observed in the 2D and 3D cases, where the regularity of the initial data on certain half-spaces propagates with infinite speed, to the case where the regularity of the initial data is measured on a fractional scale. To achieve this, we introduce new localization formulas that enable us to describe the regularity of the solution on a specific class of subsets in Euclidean space. This work provides insights into the regularity behavior of solutions of the ZK equation in higher dimensions and with more general initial data. [ABSTRACT FROM AUTHOR]

This study explores the novel solitary wave solutions of the perturbed Chen–Lee–Liu (CLL) equation, aiming to elucidate the physical and dynamic behaviors of pulses in optical fiber. The perturbed CLL equation is derived from the well-known Schrödinger equation and serves as an iconic model. Two analytical techniques are employed to obtain these novel solitary wave solutions. Subsequently, these solutions are subjected to objective analysis using a widely recognized semianalytical scheme to comprehend their underlying mechanisms. Multiple graphs with diverse styles are utilized to illustrate the analysis of pulse waves in optical fiber and assess the accuracy of the analysis. The scientific novelty of this research lies in providing a comprehensive explanation through a comparative analysis of our recently published results in related research papers. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial-boundary value (IBV) problem for a generalized Sasa–Satsuma equation with 3 × 3 Lax pair by Fokas unified method on the half line. Based on the analyticity and asymptotics of the eigenfunctions, the IBV problem is formulated as a Riemann–Hilbert (RH) problem. Further, the global relation among IBVs is established and the map from the Dirichlet boundary value to Neumann boundary value is obtained. [ABSTRACT FROM AUTHOR]

This paper deals with the convergence of finite volume scheme (FVS) for solving coagulation and multiple fragmentation equations having locally bounded coagulation kernel but singularity near the origin due to fragmentation rates. Thanks to the Dunford–Pettis and De La Vall é e-Poussin theorems, we establish that numerical solution is converging to the weak solution of the continuous model using a weak L 1 compactness argument. A suitable stable condition on time step is taken to achieve the result. Furthermore, when kernels are in W loc 1 , ∞ space, first-order error approximation is demonstrated for a uniform mesh. It is numerically validated by taking four test problems of coupled coagulation–fragmentation models. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the relation between the gradient-flow equations and Hamilton's equations in information geometry. By regarding the gradient-flow equations as Huygens' equations in geometric optics, we have related the gradient flows to the geodesic flows induced by the geodesic Hamiltonian in an appropriate Riemannian geometry. The original evolution parameter t in the gradient-flow equations is related to the arc-length parameter in the associated Riemannian manifold by Jacobi–Maupertuis transformation. As a by-product, the relation between the gradient-flow equation and replicator equations is found. [ABSTRACT FROM AUTHOR]

The quarkonium yield productions in the relativistic Heavy-Ion Collisions and their modifications within the Quark–Gluon Plasma (QGP) produced in the collisions have been investigated for more than four decades. Similarly, phenomenological model studies with various themes have been performed to comprehend the features of the QGP medium along with its associated particle production. This paper proposes a model that combines the effects of color screening, gluon-induced dissociation, and recombination on quarkonium production in heavy-ion collisions involving (Pb + Pb ions) at a center-of-mass energy of ( s NN ) = 5. 0 2 TeV. Unlike the Boltzmann transport rate equations of dissociation and recombination, the rate equations are decoupled and solved individually using a method involving Bateman solution ensuring the dissociation and recombination of heavy quarks within the QGP medium are accounted well. [ABSTRACT FROM AUTHOR]

In this paper, we apply classical non-uniform L1 formula and the compact difference scheme for solving linear fractional systems with Neumann boundary conditions. A novelty and simple demonstration strategy is presented on the convergence analysis in the discrete maximum norm. Moreover, based on the special properties of the resulting coefficient matrix, diagonalization technique and discrete cosine transform (DCT) are adopted to speed up the convergence rate of the proposed method. In addition, the numerical scheme is also extended to the three-dimensional (3D) case. Several numerical experiments are given to support our findings. [ABSTRACT FROM AUTHOR]

In this paper, the M-lump solutions, the periodic type, and cross-kink wave solutions are acquired. Here, the Hirota bilinear operator is employed. By utilizing the symbolic computation and employing the utilized method, the (3+1)-dimensional Jimbo–Miwa (JM) equation is investigated. Based on the Hirota bilinear form, the soliton solution and periodic wave solution to the mentioned equation, respectively, are obtained. We gained plenty of multiple collisions of lumps. Next, the periodic wave and cross-kink wave have greatly enriched the existing literature on the JM equation. Through the three-dimensional designs, contour design, density design, and two-dimensional design by using Maple, the physical features of these soliton solutions are explained all right. The forms of the attained solutions are one-lump, two-lumps, and three-lumps wave solutions. Then, a class of rogue waves-type solutions to the (3+1)-dimensional JM equation within the frame of the bilinear equation is found. These results can help us better understand interesting physical phenomena and mechanisms. [ABSTRACT FROM AUTHOR]

In this paper, linear higher-order parabolic equation is shown to define a semigroup in Morrey spaces. For higher-order diffusion equation, the associated semigroup is proved analytic and sharp semigroup estimates in Morrey scale are derived. The analysis also includes semigroups in Morrey scale related to solving linear higher-order fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In this paper, we use Padé approximations, constructed in [Kawashima and Poëls, Padé approximations for shifted functions and parametric geometry of numbers, J. Number Theory 243 (2023) 646–687] for binomial functions, to give a new upper bound for the number of the solutions of the S -unit equation in two variables. Combining explicit Padé approximants with a simple argument relying on Mahler measure as well as the local height, we refine the bound due to Evertse [On equations in S -units and the Thue-Mahler equation, Invent. Math. 75 (1984) 561–584]. [ABSTRACT FROM AUTHOR]

In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the L 2 -error bound and the L ∞ / H 2 -stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of − 1 instead of e T / 2 for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis. [ABSTRACT FROM AUTHOR]

Conservation laws are well known to be a crucial part of modeling. Considering such models with the inclusion of non-local flows is becoming increasingly important in many models. On the other hand, kinetic equations provide interesting theoretical results and numerical schemes for the usual conservation laws. Therefore, studying kinetic equations associated to conservation laws for non-local flows naturally arises and is very important. The aim of this paper is to propose kinetic models associated to conservation laws with a non-local flux in dimension d and to prove the existence of solutions for these kinetic equations. This is the very first result of this kind. In order for the paper to be as general as possible, we have highlighted the properties that a kinetic model must verify in order that the present study applies. Thus, the result can be applied to various situations. We present two sets of properties on a kinetic model and two different techniques to obtain an existence result. Finally, we present two examples of kinetic model for which our results apply, one for each set of properties. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalization of Gross Pitaevskii equation for condensate of bosons with nonextensive statistics. First, we use the non-additive methods and formalism to obtain the well-known Schrödinger equation. Using a suitable Hamiltonian for condensate phase and minimizing the free energy of the system by non-additive formalism, we work out the nonextensive Gross Pitaevskii equation. [ABSTRACT FROM AUTHOR]