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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]

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]

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 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]

We show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura group. [ABSTRACT FROM AUTHOR]

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class (m , n) ∈ C k (R) ∩ W k , 1 (R) with k ∈ N ∪ { 0 } . This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: ∂ t m (t , x) = ∂ x [ m (t , x) (u (t , x) - ∂ x u (t , x)) (u (- t , - x) + ∂ x (u (- t , - x))) ] , where m (t , x) = 1 - ∂ x 2 u (t , x) . Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class C k ∩ W k , 1 . Moreover, we derive criteria for blow-up of the local solution in this class. [ABSTRACT FROM AUTHOR]

Abstract 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.