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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]