Exact solutions of nonlinear dispersive K(m, n) model with variable coefficients

The variable-coefficient Korteweg-de Vries (KdV) equation with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condense was presented to describe the dynamics of nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. To understand the role of nonlinear dispersion in this variable-coefficient model, we introduce and study a new variable-coefficient KdV with nonlinear dispersion (called vc-K(m, n) equation). With the aid of symbolic computation, we obtain its compacton-like solutions and solitary pattern-like solutions. Moreover, we also present some conservation laws for both vc-K⁺(n, n) equation and vc-K⁻(n, n) equation.     

Recursive algorithm for the reliability of a connected-(1, 2)-or-(2, 1)-out-of-(m, n):F lattice system

The connected-(1, 2)-or-(2, 1)-out-of-(m, n):F lattice system is included by the connected-X-out-of-(m, n):F lattice system defined by Boehme et al. [Boehme, T.K., Kossow, A., Preuss, W., 1992. A generalization of consecutive-k-out-of-n:F system. IEEE Transactions on Reliability 41, 451–457]. This system fails if and only if at least one subset of connected failed components occurs which includes at least a (1, 2)-matrix (that is, a row and two columns) or a (2, 1)-matrix(that is, two rows and a column) of failed components. This system is applied to two-dimensional network problems with adjacent constraints, and various systems, for example, a supervision system, etc.     

Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.     

Composition operators from F(p, q, s) spaces to the nth weighted-type spaces on the unit disc

In this note we characterize the boundedness and compactness of the composition operator from the general function space F(p, q, s) to the nth weighted-type space on the unit disk, where the nth weighted-type space has been recently introduced by Stevo Stevi?.     

Special exact soliton solutions for the K(2, 2) equation with non-zero constant pedestal

Special exact solutions of the K(2, 2) equation, u_t + (u²)_x + (u²)_xxx = 0, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the K(2, 2) equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal lim_x→±∞u = A ≠ 0, or possesses compacton solutions only when lim_x→±∞u = 0. Mathematical analysis and numerical simulations are provided for these soliton solutions of the K(2, 2) equation.     

The equivalence of curves in SL(n, R) and its application to ruled surfaces

The generating system of the differential algebra for invariant differential polynomials with two parametric curves is obtained. Conditions for the equivalence of two parametric curves families are given. We are also proved that the generating differential invariants of two parametric curves are independent. Finally, we reduce the SL(n, R)-equivalent problem for ruled surfaces to that of parametric curves.     

An algorithm for mean residual life computation of (n − k + 1)-out-of-n systems: An application of exponentiated Weibull distribution

In this paper, we establish an algorithm for the computation of the mean residual life of a (n − k + 1)-out-of-n system in the case of independent but not necessarily identically distributed lifetimes of the components. An application for the exponentiated Weibull distribution is given to study the effect of various parameters on the mean residual life of the system. Also the relationship between the mean residual life for the system and that of its components is investigated.     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

Soliton solutions of a BBM(m, n) equation with generalized evolution

We consider a BBM(m, n) equation which is a generalization of the celebrated Benjamin-Bona-Mahony equation with generalized evolution term. By using two solitary wave ansatze in terms of sech^p(x) and tanh^p(x) functions, we find exact analytical bright and dark soliton solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The conditions of existence of solitons are presented. Note that, it is always useful and desirable to construct exact analytical solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.     

Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method

In this paper, we establish exact solutions for (2 + 1)-dimensional nonlinear evolution equations. The sine-cosine method is used to construct exact periodic and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Boussinesq, breaking soliton and BKP equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.     

Symmetry reductions and explicit solutions of a (3 + 1)-dimensional PDE

The symmetry of the (3 + 1)-dimensional partial differential equation has been derived via a direct symmetry method and proved to be infinite dimensional non-Virasoro type symmetry algebra. Many kinds of symmetry reductions have been obtained, including the (2 + 1)-dimensional ANNV equation and breaking soliton equation. And some new soliton solutions and complex solutions are obtained due to the Riccati equation method and symbolic computation.     

Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation

A set of sufficient conditions consisting of systems of linear partial differential equations is obtained which guarantees that the Wronskian determinant solves the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form. Upon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield rational solutions, solitons, negatons, positons and interaction solutions.     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

Exact coherent structures in the (2 + 1)-dimensional KdV equations

Different from the (1 + 1)-dimensional nonlinear systems, (2 + 1) or higher dimensional nonlinear systems admit more rich coherent structures. Taking (2 + 1)-dimensional Korteweg de Vries (KdV for short) equations as an example, the singular manifold method is applied to search these coherent structures in an analytical form. With the aid of symbolic computation and plot representation of Maple, some coherent structures expressed in terms of new forms, such as dromions and solitoffs, have been illustrated by means of arbitrary functions in the analytical forms. In the paper, we will show these results by changing some specific choices for three different special cases for singular variable in details.     

A (3 + 1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions

In this work, a (3 + 1)-dimensional nonlinear evolution equation is investigated. The Hirota’s bilinear method is applied to determine the necessary conditions for the complete integrability of this equation. Multiple soliton solutions are established to confirm the compatibility structure. Multiple singular soliton solutions are also derived. The resonance phenomenon does not exist for this model.     

A diversity of localized structures in a (2 + 1)-dimensional KdV equation

The singular manifold method is used to solve a (2 + 1)-dimensional KdV equation. An exact solution containing two arbitrary functions is then obtained. A diversity of localized structures, such as generalized dromions and solitoffs, is exposed by making full use of these arbitrary functions. These localized structures are illustrated by graphs.     

Periodic solutions of (2n+1)th order ordinary differential equations

The goal of this paper is to prove the existence and uniqueness of periodic solutions for a class of the odd order differential equations. Here the Leray-Schauder degree theory and the Schwarz inequality are used.