EXPONENTIAL STABILITY OF FIRST ORDER DYNAMIC EQUATION ON TIME SCALES

This paper is concerned with the stability of a first order dynamic equation on time scales. In particular,using the properties of the generalized exponential function with time scales,different kinds of sufficient conditions for the exponential stability are obtained.

Oscillation Criteria of Second Order Half Linear Delay Dynamic Equations on Time Scales

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation （r（t）g（x△（t）））△＋p（t）f（x（τ（t）））=0 on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=R, difference equations when T=N but can be applied on different types of time scales such as when T=N for q〉1 and also improve most previous results. Finally, we give some examples to illustrate our main results.

Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations

The prime objective of this paper is to explore the new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and(2+1)-dimensional breaking soliton equations via a generalized exponential rational function(GERF) method. Many different kinds of exact soliton solution are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact soliton solutions are also demonstrated by a choice of appropriate values of the free constants that aid in understanding the nonlinear complex phenomena of such equations. These exact soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular-form solitons, single solitons,double solitons, triple solitons, bell-shaped solitons, combo singular solitons, breather-type solitons,elastic interactions between triple solitons and kink waves, and elastic interactions between diverse solitons and kink waves. Because of the reduction in symbolic computation work and the additional constructed closed-form solutions, it is observed that the suggested technique is effective, robust, and straightforward. Moreover, several other types of higher-dimensional nonlinear evolution equation can be solved using the powerful GERF technique.

Depth in Bucket Recursive Trees with Variable Capacities of Buckets

We consider bucket recursive trees of sizen consisting of all buckets with variable capacities1,2,...,b and with a specifc stochastic growth rule.This model can be considered as a generalization of random recursive trees like bucket recursive trees introduced by Mahmoud and Smythe where all buckets have the same capacities.In this work,we provide a combinatorial analysis of these trees where the generating function of the total weights satisfes an autonomous frst order diferential equation.We study the depth of the largest label（i.e.,the number of edges from the root node to the node containing label n）and give a closed formula for the probability distribution.Also we prove a limit law for this quantity which is a direct application of quasi power theorem and compute its mean and variance.Our results for b=1 reduce to the previous results for random recursive trees.

On dynamical behavior for optical solitons sustained by the perturbed Chen–Lee–Liu model

This study investigates the perturbed Chen–Lee–Liu model that represents the propagation of an optical pulse in plasma and optical fiber.The generalized exponential rational function method is used for this purpose.As a result,we obtain some non-trivial solutions such as the optical singular,periodic,hyperbolic,exponential,trigonometric soliton solutions.We aim to express the pulse propagation of the generated solutions,by taking specific values for the free parameters existed in the obtained solutions.The obtained results show that the generalized exponential rational function technique is applicable,simple and effective to get the solutions of nonlinear engineering and physical problems.Moreover,the acquired solutions display rich dynamical evolutions that are important in practical applications.

Abundant closed-form wave solutions and dynamical structures of soliton solutions to the (3+1)-dimensional BLMP equation in mathematical physics

The physical principles of natural occurrences are frequently examined using nonlinear evolution equa-tions(NLEEs).Nonlinear equations are intensively investigated in mathematical physics,ocean physics,scientific applications,and marine engineering.This paper investigates the Boiti-Leon-Manna-Pempinelli(BLMP)equation in(3+1)-dimensions,which describes fluid propagation and can be considered as a non-linear complex physical model for incompressible fluids in plasma physics.This four-dimensional BLMP equation is certainly a dynamical nonlinear evolution equation in real-world applications.Here,we im-plement the generalized exponential rational function(GERF)method and the generalized Kudryashov method to obtain the exact closed-form solutions of the considered BLMP equation and construct novel solitary wave solutions,including hyperbolic and trigonometric functions,and exponential rational func-tions with arbitrary constant parameters.These two efficient methods are applied to extracting solitary wave solutions,dark-bright solitons,singular solitons,combo singular solitons,periodic wave solutions,singular bell-shaped solitons,kink-shaped solitons,and rational form solutions.Some three-dimensional graphics of obtained exact analytic solutions are presented by considering the suitable choice of involved free parameters.Eventually,the established results verify the capability,efficiency,and trustworthiness of the implemented methods.The techniques are effective,authentic,and straightforward mathematical tools for obtaining closed-form solutions to nonlinear partial differential equations(NLPDEs)arising in nonlinear sciences,plasma physics,and fluid dynamics.