The initial value problem of nonlinear KdV-Schrödinger system

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, particularly useful for modeling phenomena like solitons. nonlinear Schrödinger (NLS) models dynamics narrow-bandwidth wave packets consisting short waves. describing many physical systems, including Bose-Einstein condensates, optical fibers, water A system couples KdV NLS equations can interaction This combines strengths models. described by affect behavior equation, while turn Such coupled has been studied extensively over last few decades, led to important insights systems. paper considers existence local solutions Cauchy problem KdV-Schrödinger on basis literature (Bernard D, Nghiem V N, Benjamin L S <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1088/1751-8113/49/41/415501">2016 <i>J. Phys. A: Math. Theor.</i> <b>49</b> 415501</ext-link>), also gives space solutions.