The Landau–Lifshitz–Gilbert (LLG) equation is a fascinating nonlinear evolution equation both from mathematical and physical points of view. It is related to the dynamics of several important physical systems such as ferromagnets, vortex filaments, moving space curves, etc. and has intimate connections with many of the well-known integrable soliton equations, including nonlinear Schrödinger and sine-Gordon equations. It can admit very many dynamical structures including spin waves, elliptic function waves, solitons, dromions, vortices, spatio-temporal patterns, chaos, etc. depending on the physical and spin dimensions and the nature of interactions. An exciting recent development is that the spin torque effect in nanoferromagnets is described by a generalization of the LLG equation that forms a basic dynamical equation in the field of spintronics. This article will briefly review these developments as a tribute to Robin Bullough who was a great admirer of the LLG equation.

We present the exact bright one-soliton and two-soliton solutions of the integrable three coupled nonlinear Schrödinger equations (3-CNLS) by using the Hirota method, and then obtain them for the general $N$-coupled nonlinear Schrödinger equations ( $N$-CNLS). It is pointed out that the underlying solitons undergo inelastic (shape changing) collisions due to intensity redistribution among the modes. We also analyze the various possibilities and conditions for such collisions to occur. Further, we report the significant fact that the various partially coherent solitons discussed in the literature are special cases of the higher order bright soliton solutions of the $N$-CNLS equations.

By constructing the general six-parameter bright two-soliton solution of the integrable coupled nonlinear Schrödinger equation (Manakov model) using the Hirota method, we find that the solitons exhibit certain novel inelastic collision properties, which have not been observed in any other $\mathrm{}(1+1)\mathrm{}$-dimensional soliton system so far. In particular, we identify the exciting possibility of switching solitons between modes by changing the phase. However, the standard elastic collision property of solitons is regained with specific choices of parameters.

The integrability aspects of a classical one‐dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a⁴)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher‐order generalized nonlinear Schrödinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower‐order continuum isotropic spin chain, is identified by carrying out a Painlevé singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.