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| On the Solution of the Cauchy Problem for a Nonstationary Fifth-Order Modified Korteweg-De Vries Equation with an Additional Term in the Class of Rapidly Decreasing Functions |
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| KeyWord:Fifth-order modified Korteweg-de Vries equation, jost solutions, scattering data, gelfand-Levitan-Marchenko integral equation |
| Author Name | Affiliation | | Umid Azaovich Hoitmetov | Department of Applied Mathematics and Mathematical Physics, Urgench State University, 14, H. Alimdjan, 220100 Urgench, Uzbekistan | | Shekhzod Kuchkarboy ugli Sobirov | Department of Applied Mathematics and Mathematical Physics, Urgench State University, 14, H. Alimdjan, 220100 Urgench, Uzbekistan | | Rukhiya Bazar kizi Atajanova | Department of Algebra and Mathematical Engineering, Urgench State University, 14, H. Alimdjan, 220100 Urgench, Uzbekistan |
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| Abstract: |
| This paper focuses on the analytical study of the Cauchy problem for the non-stationary fifth-order modified Korteweg-de Vries equation with an additional term. A distinctive feature of the mathematical model under consideration is the presence of time-dependent coefficients, and the analysis is carried out in the space of rapidly decreasing functions, which implies an exponential tendency of solutions to zero at infinity. The central result of the paper is the establishment of the scattering data dynamics for the spectral problem associated with the Dirac operator. It is shown that the evolution of these characteristics is directly determined by the behavior of the potential, which is a solution to the fifth-order modified Korteweg-de Vries equation. The paper presents a detailed example illustrating the features of the evolution of solutions within the given initial conditions, which demonstrates the universality of the proposed approach. In addition, the Cauchy problem for the loaded fifth-order modified Korteweg-de Vries equation is considered, and a detailed example is given demonstrating the nature of the change in solutions over time. |
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