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 Schrodinger 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.

The fascinating world of the Landau-Lifshitz-Gilbert equation: an overview

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 Schr\"odinger 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 which 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.

The Fascinating World of Landau-Lifshitz-Gilbert Equation: An Overview

In this paper, we present a generalized unified method for finding multiwave solutions of the time-fractional (2+1)-dimensional Nizhnik–Novikov–Veselov equations. The fractional derivatives are described in the modified Riemann–Liouville sense. The fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann–Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation. Multiauxiliary equations have been introduced in this method to obtain not only multisoliton solutions but also multiperiodic or multielliptic solutions. It is shown that the considered method is very effective and convenient for solving wide classes of nonlinear partial differential equations of fractional order.

Multiwave solutions of time-fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations

Deformation rogue wave as exact solution of the (2+1)-dimensional Korteweg–de Vries (KdV) equation is obtained via the bilinear method. It is localized in both time and space and is derived by the interaction between lump soliton and a pair of resonance stripe solitons. In contrast to the general method to get the rogue wave, we mainly combine the positive quadratic function and the hyperbolic cosine function, and then the lump soliton can be evolved rogue wave. Under the small perturbation of parameter, rich dynamic phenomena are depicted both theoretically and graphically so as to understand the property of (2+1)-dimensional KdV equation deeply. In general terms, these deformations mainly have three types: two rogue waves, one rogue wave or no rogue wave.

Deformation rogue wave to the (2+1)-dimensional KdV equation

Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schr\"odinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schr\"odinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schr\"odinger--Hirota--Maxwell--Bloch equations, along with their Lax pairs.

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Integrable (2+1)-Dimensional Spin Models with Self-Consistent Potentials

In this paper, using a novel approach involving the truncated Laurent expansion in the Painleve analysis of the (2 + 1) dimensional K-dV equation, we have trilinearized the evolution equation and obtained rather general classes of solutions in terms of arbitrary functions. The highlight of this method is that it allows us to construct generalized periodic structures corresponding to different manifolds in terms of Jacobian elliptic functions, and the exponentially decaying dromions turn out to be special cases of these solutions. We have also constructed multi-elliptic function solutions and multi-dromions and analysed their interactions. The analysis is also extended to the case of generalized Nizhnik–Novikov–Veselov (NNV) equation, which is also trilinearized and general class of solutions obtained.

Trilinearization and localized coherent structures and periodic solutions for the (2 + 1) dimensional K-dV and NNV equations

In this paper, we investigate the (2+1)-dimensional long wave–short wave resonance interaction (LSRI) equation and show that it possess the Painleve property. We then solve the LSRI equation using Painleve truncation approach through which we are able to construct solution in terms of three arbitrary functions. Utilizing the arbitrary functions present in the solution, we have generated a wide class of elliptic function periodic wave solutions and exponentially localized solutions, such as dromions, multidromions, instantons, multi-instantons and bounded solitary wave solutions.

Periodic and localized solutions of the long wave-short wave resonance interaction equation

In this paper, using a novel approach involving the truncated Laurent expansion in the Painlev\'e analysis of the (2+1) dimensional K-dV equation, we have trilinearized the evolution equation and obtained rather general classes of solutions in terms of arbitrary functions. The highlight of this method is that it allows us to construct generalized periodic structures corresponding to different manifolds in terms of Jacobian elliptic functions, and the exponentially decaying dromions turn out to be special cases of these solutions. We have also constructed multi-elliptic function solutions and multi-dromions and analysed their interactions. The analysis is also extended to the case of generalized Nizhnik-Novikov-Veselov (NNV) equation, which is also trilinearized and general class of solutions obtained.

Trilinearization and Localized Coherent Structures and Periodic Solutions for the (2+1) dimensional K-dV and NNV equations

The Mel'nikov equation is a (2+1) dimensional nonlinear evolution equation admitting boomeron type solutions. In this paper, after showing that it satisfies the Painleve property, we obtain exponentially localized dromion type solutions from the bilinearized version which have not been reported so far. We also obtain more general dromion type solutions with spatially varying amplitude as well as induced multi-dromion solutions.

Exponentially localized solutions of Mel'nikov equation

In this communication, we investigate the two-component long-wave‐shortwave resonance interaction equation and show that it admits the Painlev´ e property. We then suitably exploit the recently developed truncated Painlev´ e

http://iopscience.iop.org/article/10.1088/1751-8113/42/10/102002/pdf

C. Senthil Kumar

Papers

Trilinearization and localized coherent structures and periodic solutions for the (2 + 1) dimensional K-dV and NNV equations

Periodic and localized solutions of the long wave-short wave resonance interaction equation

Trilinearization and Localized Coherent Structures and Periodic Solutions for the (2+1) dimensional K-dV and NNV equations

Exponentially localized solutions of Mel'nikov equation

The collision of multimode dromions and a firewall in the two-component long-wave–short-wave resonance interaction equation