Narrow Results

The goal of this paper is to present new results on the general structure of fractional Lévy–Meixner noise functionals. More precisely, we construct and emphasize certain properties of a new Appell polynomials associated with the infinite-dimensional Fractional Lévy–Meixner noise measure. Furthermore, we explore the relationship between new polynomials and the orthogonal Bell polynomials, which has also been examined and discussed.

For a smooth, irreducible projective surface S over ℂ, the number of r-nodal curves in an ample linear system (where is a line bundle on S) can be expressed using the rth Bell polynomial P_r in universal functions a_i, 1 ≤ i ≤ r, of (S, ), which are ℤ-linear polynomials in the four Chern numbers of S and . We use this result to establish a proof of the classical shape conjectures of Di Francesco–Itzykson and Göttsche governing node polynomials in the case of ℙ². We also give a recursive procedure which provides the -term of the polynomials a_i.

In this paper, the integrability of a generalized ($2+1$)-dimensional equation is investigated by using the Bell polynomials and Hirota bilinear method. By appropriate transformation, the bilinear equation is constructed. Some soliton solutions are yielded via the Hirota bilinear method. Then the evolution and collision of the soliton solutions are discussed. Starting from the two-field condition, the bilinear Bäcklund transformation is derived. Linearization of the ${𝒴}$-polynomials-typed Bäcklund transformation leads to the Lax pair. Besides, infinite conservation laws are derived by constructing the Riccati-type equation and devergence-type equation.

The temporal-second-order KdV equation, which describes the propagation of two wave modes with different phase velocities and same dispersion relation, nonlinearity and dispersion parameters are investigated. The similarity reductions and new exact solutions are obtained via the Kudryashov method and a new version of Kudryashov method. Furthermore, the conservation laws are derived using the new conservation theorem. The bilinear forms and bilinear Bäcklund transformation of the temporal-second-order KdV equation are derived through the binary Bell polynomial. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota method. The characteristics and interaction of the solitons are discussed graphically. We discuss the effect of the phase velocities $c_1$ and $c_2$ and the parameters of nonlinearity $a_1$ and $a_2$ on the soliton amplitudes and velocities. Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the temporal-second-order KdV equation.

Under investigation in this paper is a (2+1)-dimensional B-type Kadomtsev–Petviashvili equation for the shallow water wave in a fluid or electrostatic wave potential in a plasma. Bilinear form, Bäcklund transformation and Lax pair are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota’s method. Propagation and interaction of the solitons are illustrated graphically: (i) Through the asymptotic analysis, elastic and inelastic interactions between the two solitons are discussed analytically and graphically, respectively. The elastic interaction, amplitudes, velocities and shapes of the two solitons remain unchanged except for a phase shift. However, in the area of the inelastic interaction, amplitudes of the two solitons have a linear superposition. (ii) Elastic interactions among the three solitons indicate that the properties of the elastic interactions among the three solitons are similar to those between the two solitons. Moreover, oblique and overtaking interactions between the two solitons are displayed. Oblique interactions among the three solitons and interactions among the two parallel solitons and a single one are presented as well. (iii) Inelastic–elastic interactions imply that the interaction between the inelastic region and another one is elastic.

For the (3+1)-dimensional nonlinear evolution equation in a plasma, crystal or fluid, the Pfaffian solutions are obtained via the Bell polynomials, symbolic computation and Hirota method. Amplitudes and velocities of the solitons are discussed, respectively, as well as the conditions on whether the collisions are overtaking or head-on in the fluid/plasma/crystal. If the product of four wave numbers is greater than 0, collisions are overtaking, or else, head-on. For the (2+1)-dimensional Sawada–Kotera equation in a liquid, we discuss that amplitudes and velocities of the solitons, as well as the conditions of solitonic collisions. Hereby, there only exist the overtaking collisions between the solitons in such a liquid because the sign of $l_1+l_2$ is the same as $l_1^3+l_2^3$, where $l_1$ and $l_2$ are the wave numbers in the liquid. Figures showing the overtaking and head-on collisions for the two and three solitons in the fluid/plasma/crystal are also presented.

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

In this paper, the (3+1)-dimensional Boussinesq equation which can describe the propagation of gravity waves on the surface of water is investigated. Using the Bell polynomials, the bilinear form of the (3+1)-dimensional Boussinesq equation is obtained and the lump soliton solutions for the equation are derived by means of the quadratic function method. As an important integrable property, the Bäcklund transformation for the (3+1)-dimensional Boussinesq equation is constructed by the Bell polynomials considering the constraints on the derivatives with respect to spatial and temporal variables. Through the relationship between the Bell polynomials and the Hirota bilinear operators, the bilinear Bäcklund transformation for the (3+1)-dimensional Boussinesq equation is given.

In this paper, the (3+1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation is discussed, which can be used to describe certain characteristics of soliton in a nonlinear media with weak dispersion. By using the virtue of Bell polynomial, we construct the exact bilinear formalism and soliton wave of the equation, respectively. We also analyze its stability analysis. Moreover, based on the resulting bilinear formalism, we obtain its rouge wave solutions with a direct method. Finally, we also discuss the interaction phenomena between solitary wave solutions and rogue wave solutions. It is hoped that our results can be used to enrich the dynamics of the (3+1)-dimensional nonlinear wave fields.

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faà di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss Möbius inversion in certain Hopf algebras built from Bell polynomials.

The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include recurrences in terms of Bell polynomials evaluated at values of the Bessel zeta function. A probabilistic version of an identity of Euler yields additional recurrences. Connections to the umbral formalism on Bessel functions introduced by Cholewinski are established.

For any $m,n\in \mathbb{N}$ we first give new proofs for the following well-known combinatorial identities

The generalized Somos recurrence is defined by $g_{0,t}=1,g_{n,t}=n{g}_{n-1,t}^t(n\ge 1).$ In the present paper, for fixed $t>1$, we study the following asymptotic series $g_{n,t}\sim \frac{{\sigma }_t^{t^n}}{n^{1/(t-1)}}{(1+{\sum }_{k=1}^{\infty }\frac{a_k(t)}{n^k})}^{-1}$ as $n\to \infty$. By using the Bell polynomials and the cycle indicator polynomials of the symmetric group, recurrence relations and a general explicit expression for the coefficients $a_k(t)$ are developed. It is also shown that these coefficients are closely related to the Apostol–Bernoulli numbers. Furthermore, an asymptotic estimation for the coefficients $a_k(t)$ is given as $k\to \infty$.