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In this paper, we get the invariant domain of 1D non-isentropic gas dynamics equations. First of all, we construct the elementary waves with the characteristic analysis method. According to the characteristic of elementary waves, we divide the u–p plane into five areas. By analyzing the structure of Riemann solutions in each area, we find a new convex bounded domain where if the Riemann data belong to the domain, then the Riemann solutions also belong to the domain. Moreover, it is used to prove the uniform boundedness of approximate solutions built by the difference scheme, so it is the basis for the Riemann problem to be applied to the Cauchy problem.

We analyze some polynomial solutions to the Schrodinger equation with a central-field potential derived recently by means of the so-called extended Nikiforov–Uvarov method. We show that this approach does not produce the spectrum of the non-relativistic quantum-mechanical model but merely some eigenvalues for particular values of the model parameters. In addition, the approach just mentioned appears to overlook the restrictions that reveal the problem as conditionally solvable.

We show that intertwining operators for the discrete Fourier transform form a cubic algebra $\mathcal{C}_q$ with $q$ a root of unity. This algebra is intimately related to the two other well-known realizations of the cubic algebra: the Askey-Wilson algebra and the Askey-Wilson-Heun algebra., Comment: 9 pages, 22 references

We introduce an equation defined on a multi-dimensional lattice, which can be considered as an extension to the coprimeness-preserving discrete KdV like equation in our previous paper. The equation is also interpreted as a higher-dimensional analogue of the Hietarinta-Viallet equation, which is famous for its singularity confining property while having an exponential degree growth. As the main theorem we prove the Laurent and the irreducibility properties of the equation in its "tau-function" form. From the theorem the coprimeness of the equation follows. In Appendix we review the coprimeness-preserving discrete KdV like equation whichis a base equation for our main system and prove the properties such as the coprimeness., 22 pages

In this paper, we prove the existence of at least two nontrivial solutions for a class of quasilinear problems with two non-negative and continuous potentials. Thanks to the geometries of these potentials, we are able to prove compact embeddings in some weighted Sobolev spaces, and by a minimization argument, we find a positive and a nodal (or sign-changing) (weak) solution with two nodal domains or that changes the sign exactly once in RN for such problems. The nonlinearity in this problem satisfies suitable growth and monotonicity conditions, which allow this result to complement the classical results due to Liu, Wang, and Wang [Commun. Partial Differ. Equations 29, 879–901 (2004)].

In this paper, we introduce twisted Rota-Baxter operators on Lie algebras as an operator analogue of twisted r-matrices. We construct a suitable $L_\infty$-algebra whose Maurer-Cartan elements are given by twisted Rota-Baxter operators. This allows us to define cohomology of a twisted Rota-Baxter operator. This cohomology can be seen as the Chevalley-Eilenberg cohomology of a certain Lie algebra with coefficients in a suitable representation. We study deformations of twisted Rota-Baxter operators from cohomological points of view. Some applications are given to Reynolds operators and twisted r-matrices. Next, we introduce a new algebraic structure, called NS-Lie algebras, that are related to twisted Rota-Baxter operators in the same way pre-Lie algebras are related to Rota-Baxter operators. We end this paper by considering twisted generalized complex structures on modules over Lie algebras., Comment: Title has been changed slightly and some minor revision has been done. (Final version)

In this article, we give the definition and additional symmetries of the multi-component supersymmetric Kadomtsev–Petviashvili (KP) (MSKP) hierarchy, the multi-component supersymmetric B type KP hierarchy, and the multi-component supersymmetric C type KP hierarchy. The additional flows of the MSKP hierarchy constitute ⨂SW1+∞ Lie algebra. Due to the B and C [B(C)] type reduction of the multi-component supersymmetric B and C type KP hierarchies, these additional flows form the B(C) type ⨂SW1+∞ Lie algebra.

In this paper, we consider the initial value problem for the mixed Schrodinger equation. For the Schwartz initial data q0(x)∈S(R), by defining a general analytical domain and two reflection coefficients, we ever found an unified long-time asymptotic formula via the Deift–Zhou nonlinear steepest descent method. In this paper, under essentially minimal regularity assumptions on initial data in a much weak weighted Sobolev space q0(x)∈H2,2(R), we apply the ∂ steepest descent method to obtain long-time asymptotics for the mixed Schrodinger equation. In the asymptotic expression, the leading order term O(t−1/2) comes from the dispersive part qt + iqxx and the error order O(t−3/4) comes from a ∂ equation.

In this paper, we investigate the existence of a mild solution and exponential stability for a class of second-order impulsive fractional neutral stochastic differential equations (IFNSDEs) driven by fractional Brownian motion in a real and separable Hilbert space together with the semigroup of a bounded linear operator and stochastic settings. Some meaningful sufficient conditions are derived, which generalize and enhance some existing results. Finally, to show the efficacy of our results, a numerical example is provided.

This paper is concerned with the stationary approximations of the stochastic wave equations defined on Rn with critical exponents. For a class of nonlinear diffusion terms, we prove the existence and uniqueness of tempered pullback attractors of the random wave equations driven by a stationary process as an approximation to the white noise. For a linear multiplicative noise, we prove the upper semicontinuity of these attractors as the step size of the approximations approaches zero. The asymptotic compactness of the solutions on Rn is established by the idea of energy equations and the uniform estimates on the tails of the solutions.

The ∂-dressing method is developed to study the three-component coupled Hirota (tcCH) equations. We first start from a ∂-problem and construct a new spectral problem. Based on the recursive operator, we successfully derive the tcCH hierarchy associated with the given spectral problem. In addition, the soliton solutions of the tcCH equations are first obtained via determining the spectral transform matrix in the ∂-problem. Finally, one-, two-, and three-soliton solutions are analyzed to discuss the dynamic phenomena of the tcCH equations. It is remarked that the interaction between solitons depends on whether the characteristic lines intersect.

In this paper, we present the globally conservative solutions to the Cauchy problem for the modified Camassa–Holm (MOCH) equation. First, we transform the equation into an equivalent semi-linear system under new variables. Second, according to the standard ordinary differential equation theory with the aid of the conservation law, we give the global solutions of the semi-linear system. Finally, returning to the original variables, we obtain the globally conservative solutions to the MOCH equation.

This paper is devoted to studying the generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation, which includes the Camassa–Holm equation, the generalized Camassa–Holm equation, the Novikov equation, the Fokas–Olver–Rosenau–Qiao/modified Camassa–Holm equation, and the Fokas–Qiao–Xia–Li/Camassa–Holm-modified Camassa–Holm equation as its special case. We first show that the gFQXL/gCH-mCH equation is locally well-posed in Besov spaces. Then, we present a blow-up criterion and a precise blow-up scenario, and utilizing the fine structure and the H1-norm conservation, we establish a finite time blow-up result with respect to the initial data. Finally, peakons are derived in an explicit formula.

From the Lax system on a time-space scale, we derive Sine-Gordon, Burgers, and some other nonlinear equations. From the simplified two- and three-dimensional Lax systems on a time-space scale with variable nonvanishing graininess, we derive Korteweg–de Vries, Boussinesq, Krichever–Novikov, Hirota–Miwa, and other nonlinear equations on a time-space scale with soliton solutions. We also construct multi-soliton solutions for some equations by the direct method using the exponential functions on a time-space scale.

A two-component-two-dimensional coupled with one-component-three-dimensional (2C2Dcw1C3D) flow may also be called a real Schur flow (RSF), as its velocity gradient is uniformly of real Schur form, the latter being the intrinsic local property of any general flows. The thermodynamic and `vortic' fine structures of RSF are exposed and, in particular, the complete set of equations governing a (viscous and/or driven) 2C2Dcw1C3D flow are derived. The Lie invariances of the decomposed vorticity 2-forms of RSFs in $d$-dimensional Euclidean space $\mathbb{E}^d$ for any interger $d\ge 3$ are also proven, and many Lie-invariant fine results, such as those of the combinations of the entropic and vortic quantities, including the invariances of the decomposed Ertel potential vorticity (and their multiplications by any interger powers of entropy) 3-forms, then follow.

In this paper, we consider the Wong–Zakai approximations of a non-autonomous stochastic FitzHugh–Nagumo system driven by a multiplicative white noise with an arbitrary intensity. The convergence of solutions of the path-wise deterministic system to that of the corresponding stochastic system is established in higher regular spaces by means of a new iteration technique and an optimal multiplier at different stages. Furthermore, we prove that the random attractor of the path-wise deterministic system converges to that of the non-autonomous stochastic FitzHugh–Nagumo system in higher regular spaces when the size of approximation vanishes, with much looser conditions on the nonlinearity.

In this paper, we investigate the reachability of linear and non-linear systems in the sense of the ψ-Hilfer pseudo-fractional derivative in g-calculus by means of the Mittag–Leffler functions (one and two parameters). In this sense, two numerical examples are discussed in order to elucidate the investigated results.

Galilean $W_3$ vertex operator algebra $\mathcal GW_3(c_L,c_M)$ is constructed as a universal enveloping vertex algebra of certain non-linear Lie conformal algebra. It is proved that this algebra is simple by using determinant formula of the vacuum module. Reducibility criterion for Verma modules is given, and the existence of subsingular vectors demonstrated. Free field realisation of $\mathcal GW_3(c_L,c_M)$ and its highest weight modules is obtained within a rank 4 lattice VOA., 24 pages

Our purpose in this paper is the generalization of the notions of quadratic and symplectic structures to the case of 3-(Hom)-$\rho$-Lie algebras. We describe some properties of them by expressing the related lemmas and theorems. Also, we introduce the concept of 3-pre-(Hom)-$\rho$-Lie algebras and define their representation., Comment: 24 pages

In this paper, we proceed with the analysis started in \cite{bib:braga-mor-souza} and, using the Renormalization Group method, we obtain logarithmic corrections to the decay of solutions for a class of nonlinear integral equations whenever the nonlinearities are classified as marginal in the Renormalization Group sense., 20 pages

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists $\tau \in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1-\tau}$ and the nonzero entries are iid random variables with mean zero and slightly more than four finite moments, then the limiting empirical eigenvalue distribution of $X$, when properly normalized, converges in probability to the uniform distribution on the unit disk in the complex plane. The key technical result is a least singular value bound for shifted random band block matrices with genuinely sublinear bandwidth, which improves on a result of Cook in the band matrix setting., Comment: 31 pages, 4 figures; minor corrections and updated references

International audience; The generic Heun operator of Lie type is identified as a certain BC-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We also show that these Bethe roots are intimately associated with the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of O(3) and the computation of the entanglement entropy for free Fermions on the Krawtchouk chain.

We analyze a new space–time fractional diffusion equation encompassing different diffusion processes in d-dimensions. The first-order time derivative is replaced with a time derivative of the Caputo type of arbitrary order β; the spatial-fractional operator of Riesz–Feller or Riesz–Weyl type is replaced with its extension to d-dimensions, defined by means of an extended Fourier transform. The mathematical problem with the spatial-fractional operator proposed here is formulated to tackle anomalous diffusion in heterogeneous media (fractal structures) and incorporating power-law distributions. A formal solution is proposed using the Green function method which, for appropriate initial and boundary conditions, can be expressed in terms of the generalized H-function of Fox—a typical track of anomalous diffusive processes. These mathematical tools provide a new powerful framework to model anomalous diffusion and relaxation problems in heterogeneous media.

$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $\beta$-Hermite, $\beta$-Laguerre, and $\beta$-Jacobi ensembles. For fixed $N$ there exist associated weak limit theorems (WLTs) in the freezing regime $\beta\to\infty$ in the $\beta$-Hermite and $\beta$-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices $\Sigma_N$ in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed $\Sigma_N^{-1}$ with formulas for the eigenvalues and eigenvectors of $\Sigma_N^{-1}$ and thus of $\Sigma_N$. In the present paper we use these data and the theory of finite dual orthogonal polynomials of de Boor and Saff to derive formulas for $\Sigma_N$ from $\Sigma_N^{-1}$ where, for $\beta$-Hermite and $\beta$-Laguerre ensembles, our formulas are simpler than those of Dumitriu and Edelman. We use these polynomials to derive asymptotic results for the soft edge in the freezing regime for $N\to\infty$ in terms of the Airy function. For $\beta$-Hermite ensembles, our limit expressions are different from those of Dumitriu and Edelman., Comment: 32 pages, made small improvements and added references

The one-dimensional shallow water equations in Eulerian coordinates are considered. Relations between symmetries and conservation laws for the potential form of the equations and symmetries and conservation laws in Eulerian coordinates are shown. An invariant difference scheme for equations in Eulerian coordinates with arbitrary bottom topography is constructed. It possesses all the finite-difference analogs of the conservation laws. Some bottom topographies require moving meshes in Eulerian coordinates, which are stationary meshes in mass Lagrangian coordinates. The developed invariant conservative difference schemes are verified numerically using examples of flow with various bottom topographies.

In this paper, free Bosons are used to study some integrable properties of the CKP hierarchy, from the aspects of tau functions. Firstly, the modified CKP hierarchy is constructed by using free Bosons and the corresponding Lax structure is given. Then the constrained CKP hierarchy is found to be related with the modified CKP hierarchy, and the corresponding solutions are derived by using free Bosons. Next by using the relations between the Darboux transformations and the squared eigenfunction symmetries, we express the Darboux transformations of the CKP hierarchy in terms of free Bosons, by which one can better understand the essential of the CKP Darboux transformations. In particular, the additional symmetries of the CKP hierarchy can be viewed as the infinitesimal generator of the CKP Darboux transformations. Based upon these results, we finally obtain the actions of the CKP additional symmetries on the CKP tau functions constructed by free Bosons., Comment: 29 pages

We construct three classes of higher-order Darboux transformations for Schrodinger equations with quadratically energy-dependent potentials by means of generalized Wronskian determinants. Particular even-order cases reduce to the Darboux transformation for conventional (energy-independent) potentials. Our construction is based on an adaptation of the results for coupled Korteweg–de Vries equations [N. V. Ustinov and S. B. Leble, J. Math. Phys. 34, 1421 (1993)].

In this paper, we study a class of neutral time-delay control systems. The relative controllability of the linear system is given by constructing the corresponding neutral delay Gramian matrix and adopting the classical analysis idea. Next, we use Krasnoselskii’s fixed point theorem to show that the semilinear system is relatively controllable under the mild condition. Finally, we give two numerical examples to verify the theoretical results.

We formulate the the generalized Forchheimer equations for the three-dimensional fluid flows in rotating porous media. By implicitly solving the momentum in terms of the pressure's gradient, we derive a degenerate parabolic equation for the density in the case of slightly compressible fluids and study its corresponding initial, boundary value problem. We investigate the nonlinear structure of the parabolic equation. The maximum principle is proved and used to obtain the maximum estimates for the solution. Various estimates are established for the solution's gradient, in the Lebesgue norms of any order, in terms of the initial and boundary data. All estimates contain explicit dependence on key physical parameters including the angular speed., Comment: 48 pages, there has been a slight change in the title and minor improvements in the presentation with this replacement

In this work, we introduce the Z3-graded differential algebra, denoted by Ω(GLq(2)), treated as the Z3-graded quantum de Rham complex of Z3-graded quantum group GLq(2). In this sense, we construct left-covariant differential calculi on the Z3-graded quantum group GLq(2).

We analyze the set ${\cal A}_N^Q$ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an $N$-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set ${\cal A}_N^Q$ is shown to be log--convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that for mixed Weyl channels the hyper-decoherence commutes with the dynamics, so that decohering a quantum accessible channel we obtain a bistochastic matrix form the set ${\cal A}_N^C$ of classical maps accessible by a semigroup. Focusing on $3$-level systems we investigate the geometry of the sets of quantum accessible maps, its classical counterpart and the support of their spectra. We demonstrate that the set ${\cal A}_3^Q$ is not included in the set ${\cal U}^Q_3$ of quantum unistochastic channels, although an analogous relation holds for $N=2$. The set of transition matrices obtained by hyper-decoherence of unistochastic channels of order $N\ge 3$ is shown to be larger than the set of unistochastic matrices of this order, and yields a motivation to introduce the larger sets of $k$-unistochastic matrices., Comment: 33 pages, 7 figures