The sine-Gordon (SG), Korteweg-de Vries (KDV) and nonlinear Schrodinger (NLS) integrable systems are regarded as universal models of nonlinear phenomena. They can describe, at leading order, several nonlinear systems and can be integrated using the inverse scattering transform. Soliton solutions and infinitely many conserved charges are distinguishing features of these models. However, some non-linear models with important physical applications and solitary wave solutions are not integrable. Nonlinear evolution equations arise in several branches of non-linear physics, such as fluids, gravity, material science, particle physics, high-Tc superconductivity, topological quantum computation, etc. Some important methods and techniques for dealing with general nonlinear evolution systems have been continuously introduced. Recently, using analytical and numerical methods, the quasi-integrability concept has been introduced in order to deal with certain deformations of integrable systems.

The theory of integrable systems is a multidisciplinary subject, embracing algebraic, geometric and analytic approaches. In addition, some numerical simulation techniques turned out to be useful tools to uncover the soliton phenomena. Besides the inherent mathematical beauty of this theory, with its many connections to mathematics, physics and other nonlinear sciences, much of the interest is motivated by the several applications of those equations and their quasi-integrable deformations. In this Research Topic we pursue to reflect this two-fold interest. First, we seek to focus on algebraic aspects of integrable systems and their quasi-integrable deformations, in particular on anomalous zero-curvature, Riccati-type pseudopotential approaches and generalized local and non-local symmetries. In addition, the stability of the solitary waves involved deserve careful treatments in these developments. Our aim is to invite papers that search for new numerical techniques, such as pseudo-spectral, time-splitting, relaxation and related methods which are useful in the simulation of soliton phenomena. Moreover, we seek for contributions that study the potential applications of solitons equations arising in many non-linear phenomena.

Areas covered by this Research Topic include, but are not limited to:
- Nonlinear Physics
- Modern theory of fluid dynamics
- Integrable and quasi-integrable systems
- Nonlinear integro-differential systems
- Phenomenological description of nonlinear evolution systems
- Stability of solitary waves
- Riccati-type pseudo-potentials
- Anomalous Lax pair and zero curvature representations
- Numerical and analytical methods

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