Existence of solitary wave solutions for long wave-short wave interaction equations

Sürekli bir ortamda yayılan iki kısa ve bir uzun dalganın rezonans etkileşimi bir boyutlu üç kuple uzun dalga-kısa dalga (LSI) etkileşim denklemleri ile temsil edilmektedir. Kısa dalgaların aynı grup hızına sahip olması ve bu grup hızının uzun dalganın faz hızına eşit olması rezonans durumunu oluşturmaktadır. LSI denklemleri yüzey su dalgalarının etkileşimi, elastik bir ortamda yayılan iç elastik dalgaların rezonans etkileşimi gibi fiziksel olayları tanımlayan denklemler olarak türetilmiştir. Bu çalışmada LSI denklem sisteminin yalnız dalga çözümlerinin varlığı ispat edilmiştir. İspat esas olarak kübik nonlineerliğe sahip tek bileşenli nonlineer Schrödinger (NLS) denklemi için önerilmiş yaklaşımın genişletilmesi üzerine inşa edilmiştir ve kısıtlamasız bir varyasyonel problemin çözümüne dayanmaktadır. Bunun için öncelikle LSI denklemlerinin yalnız dalga çözümlerinin sağlamış olduğu iki-kuple denklem sistemi elde edilmiştir. Daha sonra Euler-Lagrange denklemleri bu iki-kuple denklem sistemini veren bir  fonksiyoneli, Gagliardo-Nirenberg eşitsizliği yardımıyla tanımlanmıştır. Böylece, yalnız dalgaların varlığı problemi  fonksiyonelinin bir minimumunun varlığının gösterilmesi problemine indirgenmiştir. Fonksiyonelin minimumunun varlığını göstermek için Lieb’in kompaktlık lemması kullanılmıştır. NLS denkleminin yalnız dalga çözümlerinin varlığını göstermek amacıyla literatürde tanımlanmış olan  fonksiyoneli  uzay boyutu için geçerli iken, şimdiki çalışmada tanımlanmış olan fonksiyonel   uzay boyutunda da geçerlidir. Ayrıca, çözümlerin pozitif tanımlılığı gösterilmiş ve fonksiyoneli ile enerji fonksiyoneli arasındaki ilişki de verilmiştir. Anahtar Kelimeler: Uzun dalga-kısa dalga etkileşim denklemleri, yalnız dalgalar.Nonlinear dispersive wave equations arise in many areas of physics, such as solid mechanics, nonlinear optics and plasma. Solitary wave solutions occur as a result of the balance of dispersive and nonlinear effects. This balance makes the localized waves travel without change of form. So there is a wide interest for the behaviour of solitary wave solutions of nonlinear wave equations: existence and uniqueness, continuous dependence to initial data, stability. In the present study we are interested in establishing the existence of solitary wave solutions for a particular nonlinear dispersive wave equation. Short waves propagating in various continuous media are governed by the nonlinear Schrödinger (NLS) equation and its generalizations. In one dimensional case there exists a unique localized solitary wave solution of the NLS equation and can easily be evaluated. The existence problem in the multidimensional case has generally been investigated by using constrained or unconstrained variational methods. The constrained variational methods are based on minimization of energy functionals, which take the infimum value at the ground state solutions, under some constraints. Weinstein proved the existence problem of solitary wave solutions of the NLS equation in the multidimensional case by the help of an unconstrained variational problem. In his study a functional associated with the Gagliardo-Nirenberg inequality is defined. The solution of the Euler-Lagrange equation of is the solitary wave solution of the NLS equation. So the existence of solitary wave solutions was shown by proving the existence of a minimizer of the functional . The resonant interaction among two short wave modes with equal group speeds and one long wave mode whose phase speed is equal to the group speed of short waves is represented by the three coupled long wave-short wave interaction (LSI) equations of the form; where, is the spatial coordinate, is the time; and and are real constants. Here represents the long wave mode; and denote short wave modes. The above three-component system of long wave-short wave interaction equations describes the resonant wave propagation in various continuous media, for instance, the surface of water and a bulk elastic medium. The aim of the present study is to prove the existence of solitary wave solutions for the LSI system by using variational methods. Substitution of solitary wave solutions of the form with into the LSI system yields the coupled system In spite of a wide interest in this system and its generalizations, a few rigorous results on the existence of solutions exist up to present. In recent years there has been a large number of studies devoted to the problem of existence of solutions for various settings of this system. In these studies, the problem of existence of nontrivial solutions has been investigated extensively using variational methods based on minimization of certain energy functionals under some constraints. Here, to prove the existence of positive solutions to the above two-component system, we use a different approach based on adapting the method used by Weinstein for the NLS equation. As the one-variable functional defined by Weinstein is valid for , a new functional associated with the Gagliardo-Nirenberg inequality has been defined for . The new two-variable functional assumes the solutions of two coupled system as the critical points. And the existence of a minimizer for is proved using Lieb's compactness Lemma. The positivity of solutions and the relation between and the energy functional are also investigated. Keywords: Long wave-short wave interaction equations, existence of solitary waves