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It is shown that both the Camassa–Holm and Novikov equations are ill-posed in B p , r 1 + 1 / p (R) with (p , r) ∈ [ 1 , ∞ ] × (1 , ∞ ] in Guo et al. (J Differ Equ 266:1698–1707, 2019) and well-posed in B p , 1 1 + 1 / p (R) with p ∈ [ 1 , ∞) in Ye et al. (J Differ Equ 367: 729–748, 2023). Recently, the ill-posedness for the Camassa–Holm equation in B ∞ , 1 1 (R) has been proved in Guo et al. (J Differ Equ 327: 127–144, 2022). In this paper, we shall solve the only left an endpoint case r = 1 for the Novikov equation. More precisely, we prove the ill-posedness for the Novikov equation in B ∞ , 1 1 (R) by exhibiting the norm inflation phenomena. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of τ -functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff–Capel totally discrete Toda equations. The hypergeometric τ -functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre–Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the Camassa–Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space B p , 1 1 ∩ C 0 , 1 with p ∈ (2 , ∞ ] is discontinuous at origin. More precisely, the initial data in B p , 1 1 ∩ C 0 , 1 can guarantee that the Camassa–Holm equation has a unique local solution in W 1 , p ∩ C 0 , 1 , however, this solution is instable and can have an inflation in B p , 1 1 ∩ C 0 , 1 . [ABSTRACT FROM AUTHOR]

In a recent paper Li et al. (Z Angew Math Phys 73:52, 2022. https://doi.org/10.1007/s00033-022-01681-4) have considered a generalized nonlinear Schrödinger equation which has extensive applications in various fields of physics and engineering. After proving Liouville integrability of this equation, they investigated the phenomenon of the modulational instability for the possible reason of the formation of the rogue waves. By means of the generalized ( 2 , N - 2 )-fold Darboux transformation, authors presented several mixed localized wave solutions, such as breathers, rogue waves and semi-rational solitons for their model equation, and accurately analyzed a number of important physical quantities. It is the aim of this Comment to point out that (i) the baseband modulation instability was developed in a wrong way and (ii) one of the two different types of Taylor series expansions for solution of Lax pair used in that article for building analytical solutions, especially the one obtained with ξ j = Z does not correspond to any solution of the spectral problem (2.1) when using u 0 x , t as the seed solution. Consequently, all mixed localized solutions that involve the mentioned Taylor series are invalid. [ABSTRACT FROM AUTHOR]

In this paper, we introduce multiple skew-orthogonal polynomials and investigate their connections with classical integrable systems. By using Pfaffian techniques, we show that multiple skew-orthogonal polynomials can be expressed by multi-component Pfaffian tau-functions upon appropriate deformations. Moreover, a two-component Pfaff lattice hierarchy, which is equivalent to the Pfaff–Toda hierarchy studied by Takasaki, is obtained by considering the recurrence relations and Cauchy transforms of multiple skew-orthogonal polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the two component Degasperis–Procesi equation in critical Besov space B ∞ , 1 1 (R) . By presenting a new construction of initial data, we proved the norm inflation of the corresponding solutions in B ∞ , 1 1 (R) and hence ill-posedness. This is quite different from the local well-posedness result (Li et al. in Monatshefte für Mathematik 200:301-313, 2023) for the Degasperis–Procesi equation in critical Besov space B ∞ , 1 1 (R) due to the coupled structure of density function. [ABSTRACT FROM AUTHOR]

The Yajima–Oikawa (YO) system is an important long-wave–short-wave resonant interaction model, which can be used to describe a fascinating resonance phenomena in diverse areas, such as hydrodynamics, nonlinear optics and biophysics. In this paper, we propose a new type integrable nonlocal YO system, which can be derived from the special reduction in the two-component YO system. We show that the binary Darboux transformation is an effective method to construct not only multi-soliton solutions, but also other types of solutions for this type nonlocal integrable systems. Additionally, some novel solutions of the nonlocal YO system are obtained, and further are analyzed in detail to reveal several interesting dynamic features, such as the moving bright soliton with sudden position shift, the collision of two-breather waves. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Cauchy problem for a generalized Camassa–Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space B ∞ , 1 1 (R) . It is shown in (J. Differ. Equ., 327:127-144,2022) that the Camassa–Holm equation is ill-posed in B ∞ , 1 1 (R) , here we turn our attention to a higher-order nonlinear generalization of Camassa–Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761-781,2005). With newly constructed initial data, we get the norm inflation in the critical space B ∞ , 1 1 (R) which leads to ill-posedness. [ABSTRACT FROM AUTHOR]

This paper is concerned with the investigation of UC and BUC plane partitions based upon the fermion calculus approach. We construct generalized the vertex operators in terms of free charged fermions and neutral fermions and present the interlacing (strict) 2-partitions. Furthermore, it is showed that the generating functions of UC and BUC plane partitions can be written as product forms. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a class of nonlinear Schrödinger equation i u t = ▵ u + u p for x ∈ T d . We prove that the PDE is integrable on the space of non-negative Fourier modes, in particular that each Fourier coefficient of a solution can be explicitly solved by quadrature. Within this subspace we demonstrate a large class of (quasi)periodic solutions all with the same frequency, as well as solutions which blowup in finite time in the L 2 norm. [ABSTRACT FROM AUTHOR]

Considering the inhomogeneity of propagation media, the nonlinear integrable equations with variable coefficients have attracted much attention in recent years since they can model various physical phenomena in the real world more accurately. In this paper, a nonlinear integrable equation with time-variable coefficients in the Kadomtsev–Petviashvili hierarchy is investigated. By introducing specific variable transformation, the discussed equation is written into one differential equation with respect to the auxiliary function. This process is similar to the bilinearization in the Hirota bilinear method. Based on the perturbation approach, the N-soliton solutions are obtained, and the M-breather solutions are derived by imposing specific constraints on parameters. The lump solution is constructed by using the generalized positive quadratic function method. Besides, the long wave limit method is utilized to explore the T-lump solutions of this integrable equation. In addition, the 3D graphics of the solitons, breathers, lumps and their interaction are drawn to exhibit the physical properties of these solutions. These figures reflect that the time-variable coefficient leads to many novel dynamic behaviors, which are rather different from those for constant coefficient integrable equation. It should be pointed that the solution is the same as the general constant coefficients equations only in the case of the time-variable function as a constant. Therefore, the results obtained in this paper are more advantageous for the analysis of the complex physical applications governed by this model. [ABSTRACT FROM AUTHOR]

The paper aims to analyze the commutative property of reciprocal transformations and dimensional deformations using conservation laws. First, a geometric proof of the commutative property of reciprocal transformations is presented, based on the coordinate-free property of the exponential map. Second, it is shown that the deformation algorithm does not always keep the commutative property. Illuminating examples are provided. [ABSTRACT FROM AUTHOR]

The μ -Camassa–Holm equation and the modified μ -Camassa–Holm equation, as the nonlocal counterparts of the Camassa–Holm and modified Camassa–Holm equations, are two integrable models. Local well-posedness of the Cauchy problems for the two equations has been established in the space C ([ 0 , T) , H s (S)) ⋂ C 1 ([ 0 , T) , H s - 1 (S)) , respectively, when s > 3 / 2 and s > 5 / 2 . This paper mainly studies ill-posedness of these two equations for s < 3 / 2 . First, we show that the modified μ -Camassa–Holm equation is ill-posed due to failure of uniqueness, which is carried out by constructing a 2-peakon solution with an asymmetric peakon-antipeakon initial profile and utilizing the time-reversibility of the equation. Second, by estimating the H s -norm of peakon-antipeakon solutions as time tends to the lifespan, we investigate ill-posedness of the Cauchy problem for the μ -CH equation, respectively for 1 < s < 3 / 2 , s = 1 and s < 1 . [ABSTRACT FROM AUTHOR]

In this paper, for the extended lattice Gelfand–Dickey hierarchy, we construct its n-fold Darboux transformation and additional flows. And we prove that these flows are actually symmetries of the extended lattice Gelfand–Dickey hierarchy. Further, we show how the additional flows act on the tau function. On this basis, we generalize the extended lattice Gelfand–Dickey hierarchy to the multicomponent and noncommutative versions, and give the Lax equations, Sato equations, zero-curvature equations and other equivalent expressions of these versions. Moreover, we investigate their Darboux transformations and additional symmetries. [ABSTRACT FROM AUTHOR]

Discrete integrable systems are closely related to numerical linear algebra. An important discrete integrable system is the discrete Lotka–Volterra (dLV) system, which is a time discretization of predator–prey dynamics. Discrete time evolutions of the dLV system correspond to a sequence of LR transformations that generate matrix similarity transformations. Previously, we leveraged this property to design the so-called dLV algorithm for computing eigenvalues and singular values. In this paper, by introducing shifts of origin into the LR transformations, we propose a new shifted algorithm as a version of the dLV algorithm for convergence acceleration. The proposed algorithm is similar to the modified dLV with shift algorithm in that it is based on the LR transformations generated by the dLV system but it has the advantage that it does not require extra auxiliary variables. We present the convergence rate and numerical errors of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper is dedicated to extending the focusing/defocusing Calogero–Moser–Sutherland cubic derivative Schrödinger equations (CMSdNLS) i ∂ t u + ∂ x 2 u = ± u D + | D | | u | 2 , D = - i ∂ x , x ∈ R or x ∈ T : = R / 2 π Z , which were initially introduced in Matsuno (Phys Lett A 278(1–2):53–58, 2000; Inverse Probl 18:1101–1125, 2002; J Phys Soc Jpn 71(6):1415–1418, 2002; Inverse Prob 20(2):437–445, 2004), Abanov et al. (J Phys A 42(13): 135201, 2009), Gérard and Lenzmann (The Calogero–Moser derivative nonlinear Schrödinger equation, Communications on Pure and Applied Mathematics. ) and Badreddine (On the global well-posedness of the Calogero–Sutherland derivative nonlinear Schrödinger equation, Pure and Applied Analysis. ; Traveling waves and finite gap potentials for the Calogero–Sutherland derivative nonlinear Schrödinger equation, Annales de l’Institut Henri Poincaré, Analyse Non Linéaire. ), to a system of two matrix-valued variables, leading to the following intertwined system, i ∂ t U + ∂ x 2 U = - 1 2 U D + | D | V ∗ U - 1 2 V D + | D | U ∗ U , i ∂ t V + ∂ x 2 V = - 1 2 V D + | D | U ∗ V - 1 2 U D + | D | V ∗ V. This system enjoys a Lax pair structure, enabling the establishment of an explicit formula for general solutions on both the 1-dimensional torus and the real line. Consequently, this system can be regarded as an integrable perturbation and extension of both the linear Schrödinger equation and the CMSdNLS equations. [ABSTRACT FROM AUTHOR]

We consider the 3D Mikhalev system, u t = w x , u y = w t - u w x + w u x , which has first appeared in the context of KdV-type hierarchies. Under the reduction w = f (u) , one obtains a pair of commuting first-order equations, u t = f ′ u x , u y = (f ′ 2 - u f ′ + f) u x , which govern simple wave solutions of the Mikhalev system. In this paper we study higher-order reductions of the form w = f (u) + ϵ a (u) u x + ϵ 2 [ b 1 (u) u xx + b 2 (u) u x 2 ] + ⋯ , which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at ϵ n are assumed to be differential polynomials of degree n in the x-derivatives of u. We will view w as an (infinite) formal series in the deformation parameter ϵ . It turns out that for such a reduction to be non-trivial, the function f(u) must be quadratic, f (u) = λ u 2 , furthermore, the value of the parameter λ (which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues, λ = 1 and λ = 3 / 2 , as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of linear degeneracy of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations. [ABSTRACT FROM AUTHOR]

The Heisenberg supermagnet model is a significant supersymmetric integrable model which is the superextension of the Heisenberg ferromagnet model. This paper is concerned with the construction of the multi-component (1+1) and (2+1)-dimensional Heisenberg supermagnet models with two distinct quadratic constraints of the superspin variables. In terms of the multi-component gauge transformation, we construct their multi-component gauge equivalent counterparts, i.e., the multi-component (1+1) and (2+1)-dimensional super nonlinear Schrödinger equations and fermionic nonlinear Schrödinger equations, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we construct a symplectic bigraded Toda hierarchy which contains an symplectic deformation of the original Toda lattice hierarchy. In particular, we give the rational solutions which are expressed by the products of the symplectic Schur polynomials. [ABSTRACT FROM AUTHOR]

In this article, we investigate non-traveling wave solutions for the (2+1)-dimensional extended variable coefficients Bogoyavlenskii–Kadomtsev–Petviashvili equation with time-dependent coefficients (VC-BKP). Inspired by Shang [ 24 ] , we apply the extended three-wave method and the generalized variable separation method to the investigated problem for the first time in this article. The technique is effective, easily applicable, and reliable in solving non-traveling wave solutions. We successfully obtain forty-four exact non-traveling solutions, including double periodic solutions, kinky breather wave solution, periodic cross-kink solution and some new exact non-traveling solutions obtained firstly in this paper. These results all have a tail which gives a prediction of physical phenomenon. Moreover, we discuss the arbitrary coefficients of solutions in the real, purely imaginary and complex domains, which greatly enriches the forms of solutions. The dynamic phenomena of four types of exact solutions are demonstrated by contour, 2D and 3D graphics, which help to show their physical interpretation. [ABSTRACT FROM AUTHOR]

Researches on the nonlinear lattice equations are active, with the applications in nonlinear optics, condensed matter physics, plasma physics, etc. What we study in this paper is a three-field lattice system, which can be reduced to a modified Toda lattice system and a coupled lattice system. Based on a known Lax pair, we present an N-fold Darboux matrix, and then construct an N-fold Darboux transformation for that system, where N is a positive integer. The first three conservation laws of that system are determined via the Lax pair. Utilizing that N-fold Darboux transformation with N = 1 and 2, we obtain the one-fold solutions and two-fold solutions of that system. Those solutions can be used to describe the discrete solitons. Via the one-fold solutions, we present a combination of the kink-shaped discrete one soliton and bell-shaped discrete one soliton. Amplitude, shape and velocity of that combination remain unchanged during the propagation. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate the reverse-space–time nonlocal nonlinear Fokas–Lenells equation by the idea of Ablowitz and Musslimani. The reverse-space–time Fokas–Lenells equation, associated with a 2 × 2 matrix Lax pair, is the important integrable system, which can be reduced to the nonlocal Fokas–Lenells equation. Based on its Lax pair, we construct nonlocal version of N-fold Darboux transformation (DT) for the Fokas–Lenells equation, and obtain two kinds of soliton solutions from vanishing and plane wave backgrounds. Further some novel one-soliton and two-soliton are derived with the zero and nonzero seed solutions through complex computations, including the bright soliton, kink soliton and breather wave soliton. Moreover, various graphical analyses on the presented solutions are made to reveal the dynamic behaviors, which display the elastic interactions between two solitons and their amplitudes keeping unchanged after the interactions except for the phase shifts. It is clearly shown that these solutions have new properties which differ from ones of the classical Fokas–Lenells equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the generalized Camassa–Holm equation that containing, as its members, three integrable equations: the Camassa–Holm equation, the Degasperis–Procesi equation and the Novikov equation. We present a new and unified method to prove the sharp ill-posedness for the generalized Camassa–Holm equation in B p , ∞ s with s > max { 1 + 1 / p , 3 / 2 } and 1 ≤ p ≤ ∞ in the sense that the solution map to this equation starting from u 0 is discontinuous at t = 0 in the metric of B p , ∞ s . Our result covers and improves the previous work given in Li et al. (J Differ Equ 306:403–417, 2022), solving an open problem left in Li et al. (2022). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to discuss the Cauchy problem of integrable nonlocal (reverse-space-time) Kundu–Eckhaus (KE) equation through the Riemann–Hilbert (RH) method. Firstly, based on the zero-curvature equation, we present an integrable nonlocal KE equation and its Lax pair. Then, we discuss the properties of eigenfunctions and scattering matrix, such as analyticity, asymptotic behavior, and symmetry. Finally, for the prescribed step-like initial value: u (z , t) = o (1) , z → - ∞ and u (z , t) = R + o (1) , z → + ∞ , where R > 0 is an arbitrary constant, we consider the initial value problem of the nonlocal KE equation. The paramount techniques is the asymptotic analysis of the associated RH problem. [ABSTRACT FROM AUTHOR]

Resonant phenomena have been observed and investigated in various situations, such as plasma experiments, the maritime security and the microtubule in cell physiology. In this paper, abundant resonant behaviors are studied for the (3+1)-dimensional BKP-Boussinesq equation. We mainly discuss the resonant two- and three-soliton solutions in the (x, y)-plane and (x, z)-plane. The characteristics are given for the kink soliton waves, including expressions, maximums, minimums and velocities. The kink soliton waves in the (x, y)-plane are parallel, and the fusion or fission may occur. The kink soliton waves in the (x, z)-plane are not parallel and the resonant phenomena among them are more complicated. [ABSTRACT FROM AUTHOR]

The A ^ 2 n (2) -hierarchy can be constructed from a splitting of the Kac–Moody algebra of type A ^ 2 n (1) by an involution. By choosing certain cross section of the gauge action, we obtain the A ^ 2 n (2) -KdV hierarchy. They are the equations for geometric invariants of isotropic curve flows of type A, which gives a geometric interpretation of the soliton hierarchy. In this paper, we construct Darboux and Bäcklund transformations for the A ^ 2 n (2) -hierarchy, and use it the construct Darboux transformations for the A ^ 2 n (2) -KdV hierarchy and isotropic curve flows of type A. Moreover, explicit soliton solutions for these hierarchies are given. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin completed cycles (a generalized version of the Giacchetto–Kramer–Lewański conjecture). [ABSTRACT FROM AUTHOR]

In this paper, for supersymmetric quantum integrable spin chains with rational G l (N | M) -invariant R -matrices, we construct a coupled master T -operator which represents a generating function for two-folds commuting quantum transfer matrices. We show that the functional relations for the quantum transfer matrices are equivalent to an infinite set of Hirota bilinear equations of the modified universal character hierarchy. Also the free fermion representation of the tau function of the supersymmetric quantum two-component spin chains will be given with the help of two sets of Clifford algebras. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem to the semilinear fourth-order Schrödinger equations: i ∂ t u + ∂ x 4 u = G ∂ x k u k ≤ γ , ∂ x k u ¯ k ≤ γ , t > 0 , x ∈ R , u | t = 0 = u 0 ∈ H s (R) , (4 NLS) where γ ∈ { 1 , 2 , 3 } and the unknown function u = u (t , x) is complex valued. In this paper, we consider the nonlinearity G of the polynomial G (z) = G (z 1 , ... , z 2 (γ + 1)) : = ∑ m ≤ | α | ≤ l C α z α , for z ∈ C 2 (γ + 1) , where m , l ∈ N with 3 ≤ m ≤ l and C α ∈ C with α ∈ (N ∪ { 0 }) 2 (γ + 1) is a constant. The purpose of the present paper is to prove well-posedness of the problem (4NLS) in the lower order Sobolev space H s (R) or with more general nonlinearities than previous results. Our proof of the main results is based on the contraction mapping principle on a suitable function space employed by Pornnopparath (J Differ Equ, 265:3792–3840, 2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by Bejenaru et al. (Ann Math 173:1443–1506, 2011). Moreover we discuss scattering of global solutions and the optimality for the regularity of our well-posedness results, namely we prove that the flow map is not smooth in several cases. [ABSTRACT FROM AUTHOR]

In this paper, we first define supersymmetric Schur Q-functions and give their vertex operators realization. By means of the vertex operator, we obtain a series of non-linear partial differential equations of infinite order, called the super BKP hierarchy and the super BKP hierarchy governs the supersymmetric Schur Q-functions as the tau functions. Moreover, we prove that supersymmetric Schur Q-functions can be viewed as compound Schur Q-functions. This means that we can study the properties of supersymmetric Schur Q-functions according to Schur Q-functions, such as their applications in representation theory. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study soliton solutions for nonlocal Kundu-nonlinear Schrödinger (Kundu-NLS) equation via the Darboux transformation. The nonlocal Kundu-NLS equation can be obtained through a symmetry reduction r (x , t) = q ∗ (- x , t) . The form of N-soliton solutions for the nonlocal Kundu-NLS equation can be investigated via the one-fold and n-fold Darboux transformation. Particularly, from the Darboux transformation of the nonlocal Kundu-NLS equation, we obtain some exact solutions for the nonlocal Kundu-NLS equation with different spectral parameters and corresponding graphs are given. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized discrete Lotka-Volterra equation and explore its connections with symmetric orthogonal polynomials, Hankel determinants and convergence acceleration algorithms. Firstly, we extend the fully discrete Lotka-Volterra equation to a generalized one with a sequence of given constants { u 0 (n) } and derive its solution in terms of Hankel determinants. Then, it is shown that the discrete equation of motion is transformed into a discrete Riccati system for a discrete Stieltjes function, hence leading to a complete linearization. Besides, we obtain its Lax pair in terms of symmetric orthogonal polynomials by generalizing the Christoffel transformation for the symmetric orthogonal polynomials. Moreover, a generalization of the famous Wynn's ε-algorithm is also derived via a Miura transformation to the generalized discrete Lotka-Volterra equation. Finally, the numerical effects of this generalized ε-algorithm are discussed by applying to some linearly, logarithmically convergent sequences and some divergent series. [ABSTRACT FROM AUTHOR]