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We discuss the monic polynomials of degree n orthogonal with respect to the perturbed Gaussian weight w (z , t) = | z | α ( z 2 + t) λ e − z 2 , z ∈ R , t > 0 , α > − 1 , λ > 0 , which arises from a symmetrization of a semi-classical Laguerre weight w Lag (z , t) = z γ (z + t) ρ e − z , z ∈ R + , t > 0 , γ > − 1 , ρ > 0. The weight wLag(z) has been widely investigated in multiple-input multi-output antenna wireless communication systems in information theory. Based on the ladder operator method, two auxiliary quantities, Rn(t) and rn(t), which are related to the three-term recurrence coefficients βn(t), are defined, and we show that they satisfy coupled Riccati equations. This turns to be a particular Painlevé V (PV, for short), i.e., P V λ 2 2 , − (1 − (− 1) n α) 2 8 , − 2 n + α + 2 λ + 1 2 , − 1 2 . We also consider the quantity σ n (t) ≔ 2 t d d t ln D n (t) , which is allied to the logarithmic derivative of the Hankel determinant Dn(t). The difference and differential equations satisfied by σn(t), as well as an alternative integral representation of Dn(t), are obtained. The asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s ≔ 4nt is fixed, are established. Finally, by using the second order difference equation satisfied by the recurrence coefficients, we obtain the large n full asymptotic expansions of βn(t) with the aid of Dyson's Coulomb fluid approach. By employing these results, the second differential equations satisfied by the orthogonal polynomials will be reduced to a confluent Heun equation. [ABSTRACT FROM AUTHOR]

We discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with a singularly perturbed Gaussian weight w (z) = | z | α ⁡ exp − z 2 − t / z 2 , z ∈ R , t > 0 , α > 1. Based on the ladder operator approach, two auxiliary quantities are defined. We show that the auxiliary quantities and the recurrence coefficients satisfy some equations with the aid of three compatibility conditions, which will be used to derive the Riccati equations and Painlevé III. We show that the Hankel determinant has an integral representation involving a particular σ-form of Painlevé III and to calculate the asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s = (2n + 1 + λ)t is fixed, where λ is a parameter with λ ≔ (α ∓ 1)/2. The asymptotic behaviors of the Hankel determinant for large s and small s are obtained, and Dyson's constant is recovered here. They have generalized the results in the literature [Min et al., Nucl. Phys. B 936, 169–188 (2018)] where α = 0. By combining the Coulomb fluid method with the orthogonality principle, we obtain the asymptotic expansions of the recurrence coefficients, which are applied to derive the relationship between second order differential equations satisfied by our monic orthogonal polynomials and the double-confluent Heun equations as well as to calculate the smallest eigenvalue of the large Hankel matrices generated by the above weight. In particular, when α = t = 0, the asymptotic behavior of the smallest eigenvalue for the classical Gaussian weight exp(−z2) [Szegö, Trans. Am. Math. Soc. 40, 450–461 (1936)] is recovered. [ABSTRACT FROM AUTHOR]

In this paper, we study a multicomponent weakly interacted generalized (3 + 1)‐dimensional Kadomtsev–Petviashvili equation (MWIGKP) from which we can get many different types of equations by choosing different coefficients. Then, we deduce the Bäcklund transformation and Hirota bilinear equations of the equation. Finally, taking the two‐component case as an example, we solve the soliton solutions and the rogue wave solutions in detail. By considering the figures of the weakly coupled rogue wave solutions, we can see that the figure corresponding to first component u is eye shaped, while the figure corresponding to the second component u^ is butterfly shaped. [ABSTRACT FROM AUTHOR]

This paper aims at additional symmetries of the unextended and extended, commutative and noncommutative dispersionless Gelfand–Dickey (dGD) hierarchies. Being similar to the Lax formalism of the Gelfand–Dickey (GD) hierarchy, we construct the function ℒ and Orlov–Schulman function ℳ of the hierarchies. Meanwhile, the additional symmetry will be studied with the infinite flows of v i and τ function of the dGD hierarchy and one can find that only a part of additional flows can survive under the GD constraints with the corresponding string equation. Furthermore, we pay attention to the additional symmetries of the dispersionless extended Gelfand–Dickey (dEGD) hierarchy which has a quantum torus algebraic structure and show the flows in detail. The additional symmetry of dispersionless noncommutative Gelfand–Dickey (dNCGD) hierarchy and dispersionless extended noncommutative Gelfand–Dickey (dENCGD) hierarchy are studied. [ABSTRACT FROM AUTHOR]

In this paper, based on the Hirota and Maxwell–Bloch (H‐MB) system and its application in the theory of the femtosecond pulse propagation through an erbium doped fiber, we define two kinds of nonlocal Hirota and Maxwell–Bloch (NH‐MB) systems, namely, PT‐symmetric NH‐MB system and reverse space‐time NH‐MB system. Then, we construct the Darboux transformations of these NH‐MB systems. Meanwhile, we derive the explicit solutions by the Darboux transformations. [ABSTRACT FROM AUTHOR]

We consider two different subjects: the -deformed universal characters and the -deformed universal character hierarchy. The former are an extension of -deformed Schur polynomials, and the latter can be regarded as a generalization of the -deformed KP hierarchy. We investigate solutions of the -deformed universal character hierarchy and find that the solution can be expressed by the boson–fermion correspondence. We also study a two-component integrable system of -difference equations satisfied by the two-component universal character. [ABSTRACT FROM AUTHOR]

We use the group-based discrete moving frame method to study invariant evolutions in a -dimensional centro-affine space. We derive the induced integrable equations for invariants, which can be transformed to local and nonlocal multi-component Toda lattices under a Miura transformation, and thus establish their geometric realizations in centro-affine space. [ABSTRACT FROM AUTHOR]

In this paper, the additional symmetries and the string equation of the q -NKdV hierarchy are established basing on the Orlov–Shulman's M operator, and then some properties are also given. Next, we construct the quantum torus symmetry of the q -NKdV hierarchy and further derive the quantum torus constraints on the tau function of the q -NKdV hierarchy. Comparing to the W ∞ infinite-dimensional Lie symmetry, this quantum torus symmetry has a nice algebraic structure with double indices. Finally, we study the ghost symmetry of the multicomponent q -NKdV hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we first define a supersymmetric Gelfand–Dickey hierarchy using the Sato equation. Then we introduce some time variables and derivations involving those that from a non-abelian Lie superalgebra. Next, we construct additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the aid of the Orlov–Schulman operators which involve the above time variables and dressing operators. We give the additional flow equations of the supersymmetric Gelfand–Dickey hierarchy, and prove that the additional flows are symmetries of the supersymmetric Gelfand–Dickey hierarchy. In this way, an isomorphism between the additional symmetries of the supersymmetric Gelfand–Dickey hierarchy and the super W ∞ algebra is constructed. Finally, the B type supersymmetric Gelfand–Dickey hierarchy is constructed and we consider the additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the B type condition which is compatible with the flows of the B type supersymmetric Gelfand–Dickey hierarchy. [ABSTRACT FROM AUTHOR]

By symbolic computation and searching for the solutions of the positive quadratic functions of the related bilinear equations, two kinds of lump solutions of the (3 + 1)-dimensional weakly coupled Hirota bilinear equation are derived, and the practicability of this method is verified. Then we add an exponential function to the original positive quadratic function, and obtain a new solution of the Hirota bilinear equation. The interaction between the lump solutions and lump-kink solutions is included in the new solution. On this basis, we give the possibility of adding multiple exponential functions. Finally, we give the coupled reduced Hirota bilinear equation lump-kink solitons by combining the above two methods. In order to ensure the analyticity and reasonable localization of the block, two sets of necessary and sufficient conditions are given for the parameters involved in the solution. The local characteristics and energy distribution of bulk solution are analyzed and explained. [ABSTRACT FROM AUTHOR]

We construct a solution expressed in terms of Schur Q-functions of a strongly coupled B-type Kadomtsev-Petviashvili hierarchy. As a generalization of these functions, we introduce universal characters satisfying the bilinear equations of a new infinite-dimensional integrable system called the strongly coupled B-type universal character hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we construct weakly and strongly coupled intermediate long-wave (ILW) hierarchies which contain integrable weakly (strongly) coupled Benjamin–Ono2 equations as the primary equations. The dressing operators Φ ± of the weakly and strongly coupled ILW hierarchies satisfy the Sato equations of a weakly (strongly) coupled KP hierarchy. We found an infinite system of commuting integrals of motion related to the weakly and strongly coupled Benjamin–Ono integrable hierarchies. [ABSTRACT FROM AUTHOR]

The short pulse equation is able to describe ultra short pulse, which plays a crucial part in the field of optical fiber propagation. In this paper, we investigate a generalized complex short pulse equation and its two-component generalization. We first prove that they are Lax integrable. Subsequently, we obtain their new Lax pairs through hodograph transformation to carry out Darboux transformation, respectively. For the generalized complex short pulse equation, we provide a different Darboux matrix and verify that it is feasible, then we focus on higher-order semi-rational soliton solutions by means of generalized Darboux transformation. For the coupled generalized complex short pulse equations, we apply Darboux transformation to discuss exact solutions by choosing different seed solutions. [ABSTRACT FROM AUTHOR]