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By introducing the conception of given curvature formulation for partial differential equations, we show that the Landau–Lifshitz equation (LLE) is gauge equivalent to a nonlinear Schrodinger-like equation, which answers a question proposed by Bordag and Yanovski. As a direct consequence, explicit first-order approximate 1-soliton solutions (with respect to the interaction anisotropy) to the LLE are constructed from those to the nonlinear Schrodinger-like equation.

In this article, by means of considering a 4×4 discrete isospectral problem, and constructing a proper continuous time evolution equation, and using discrete zero curvature equation, a Blaszak–Marciniak four-field lattice hierarchy is re-derived. Thus a new matrix Lax representation for the hierarchy is obtained. From the new matrix Lax representation, we demonstrate the existence of infinitely many conservation laws for the lattice hierarchy and give the corresponding conserved densities and the associated fluxes formulaically. Thus its integrability is further confirmed.

The Sasa–Satsuma equation is an integrable high-order nonlinear Schrodinger (NLS) equation, and also is a complex modified KdV-type equation. It can describe the propagation of femtosecond pulses in optical fibers. Very recently, Ablowitz and Mussliman introduced a class of reverse space-time and reverse time nonlinear integrable equations, including the reverse space nonlocal NLS equation, the real and complex reverse space-time nonlocal mKdV, sine-Gordon, Davey–Stewartson equations, etc. So, what is nonlocal version of high-order NLS? In this paper, we introduce a reverse space-time nonlocal Sasa–Satsuma equation, i.e., a reverse space-time nonlocal high-order NLS equation, and derive its solutions with the binary Darboux transformation method. Periodic solutions, and some localized solutions, such as dark soliton, W-shaped soliton, M-shaped soliton and breather soliton of the reverse space-time nonlocal Sasa–Satsuma equation are constructed.

Stationary solitons of the fifth order singularly perturbed KdV-type equation are investigated. The second order approximate solution to fifth order singularly perturbed KdV-type equation and third...

A differential-difference Caudrey-Dodd-Gibbon-Kotera-Sawada equation is proposed. The corresponding Backlund transformation is presented. A Lax pair for it is found. A nonlinear superposition formula is also derived based on the permutability of the Backlund transformation.

Two coupled integrable hierarchies associated with a proper isospectral problem are obtained. These hierarchies possess hereditary symmetry structure but each hierarchy is not Hamiltonian.

From a proper 2×2 discrete isospectral problem, the relativistic Volterra lattice introduced by Suris and Ragnisco is rederived. So, a new Lax matrix for the relativistic Volterra lattice is given....

Some new nonlinear differential-difference integrable hierarchies associated with a properly discrete spectral problem are obtained. The special cases of the proposed differential-difference integrable hierarchies are discussed. The Ablowitz-Ladik discretization of the NLS equation, the discrete modified KdV equation, the modified Toda lattice, the relativistic Toda lattice, and some other lattice soliton equations obtained by Suris are the special examples of novel differential-difference integrable hierarcies.

A new Lax integrable hierarchy associated with a properly spectral problem is obtained. Using the trance identity method, the Hamiltonian form of the proposed Lax hierarchy is also given.