In this paper, we study the initial‐boundary‐value problem (IBVP) for coupled Korteweg–de Vries equations posed on a finite interval with nonhomogeneous boundary conditions. We overcome the requirement for stronger smooth boundary conditions in the traditional method via the Laplace transform. Our approach uses the strong Kato smoothing property and the contraction mapping principle.
The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons.The Nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature[3], and also gives the existence space of the local solutions.
In this paper, we study the initial-boundary-value problem (IBVP) for
coupled Korteweg-de Vries equations posed on a finite interval with
nonhomogeneous boundary conditions. We overcome the requirement for
stronger smooth boundary conditions in the traditional method via the
Laplace transform. Our approach uses the strong Kato smoothing property
and the contraction mapping principle.
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