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The mathematical modeling of physical phenomena often leads to a certain class of nonlinear equations known as "integrable systems". Because of their broad applicability in areas ranging from fluid dynamics and nonlinear optics to plasma physics and Bose-Einstein condensates (BECs), these systems continue to be extensively studied worldwide. Many fundamental questions remain open. The class of integrable systems includes nonlinear scalar and coupled partial differential equations (PDEs), as well as nonlinear discrete evolution equations (discrete in space–continuous time or discrete in space and time) and even fractional nonlinear systems. Among the distinguished features of integrable systems is that they admit soliton solutions, i.e., stable, localized traveling waves which preserve their shape and velocity upon interaction. Integrable systems also possess a rich and beautiful mathematical structure, most notably the fact that their initial-value problem (IVP) can be effectively linearized via the Inverse Scattering Transform (IST), as well as many other remarkable properties: an infinite number of symmetries and conserved quantities, bi-Hamiltonian structure, Bäcklund and Darboux transformations, as well as deep connections to many other branches of mathematics, e.g., algebraic and symplectic geometry, complex analysis and Riemann-Hilbert problems, Painlevé equations, Lie groups and Kac-Moody algebras, inverse problems, random matrix theory, etc.

The study of integrable systems is an active and broad field of research which has seen an astounding revival in recent years, with an unprecedented number of papers devoted to the study of soliton interactions, long-time asymptotic behavior of solutions, new classes of integrable nonlinear equations,  and exciting new applications in numerous experimental settings, from the investigation of the integrable nature of modulational instability (MI) and integrable turbulence, to soliton gases and connections to rogue waves.

The goal of this special issue of AIMS Mathematics on "Integrable Systems and Applications" is to collect important new results by well-known experts in the field and present state-of-the-art applications of these systems.

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