This article is devoted to the integration of the loaded source Korteweg-de Vries equation in the class of rapidly decreasing functions. In this work, the Cauchy problem imposed on the Korteweg-de Vries equation was solved using the inverse problem method of the Sturm-Liouville operator scattering theory. Their Yost solutions are defined and integral Levin images are obtained for them. The givens of the scattering theory were described and some of their necessary properties were given, the Gelfand-Levitan-Marchenko integral equation, which is the main integral equation of the inverse problems of the scattering theory, was derived.

Introduction

Conclusion

This article is devoted to the integration of the Korteweg-de Vries equation with an adapted source load in the class of rapidly decreasing functions. This article provides the necessary information on the exact and inverse problems of the scattering theory for the Sturm-Liouville operator, which are necessary for the following statements. First, the Yost solutions of the Sturum-Liouville operator on the entire axis are defined and integral images are obtained for them, the givens of the scattering theory are described and some of their necessary properties are given, the Gelfand-Levitan equation, which is the main integral equation of the inverse problems of the scattering theory, The Marchenco integral equation was derived. The problem of finding the solution of the Cauchy problem in the class of rapidly decreasing functions, which is applied to the Korteweg-de Vries equation with an adapted source load, is studied. In this case, the method of inverse problems of the scattering theory was used to determine the solution of the Cauchy problem imposed on the Korteweg-de Vries equation with an adapted source load in the class of rapidly decreasing functions. Equations for calculating the evolution of the Sturm-Liouville operator given by the scattering theory have been derived. The algorithm of applying the method of inverse problems of scattering theory is given. An example is shown in order to show the correctness of the obtained results.

References

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