# Article

Full entry | PDF   (0.3 MB)
Keywords:
coupled system of nonlinear Schrödinger equation and generalized IMBq; multidimensional; periodic boundary value problem; Cauchy problem; generalized local solution; classical local solution
Summary:
The existence, uniqueness and regularity of the generalized local solution and the classical local solution to the periodic boundary value problem and Cauchy problem for the multidimensional coupled system of a nonlinear complex Schrödinger equation and a generalized IMBq equation $$i\varepsilon_t+\nabla^2\varepsilon-u\varepsilon=0,$$ $$u_{tt}-\nabla^2u-a\nabla^2u_{tt}=\nabla^2f(u)+\nabla^2(|\varepsilon|^2)$$ are proved.
References:
[1] Makhankov V.G.: On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying Boussinesq's equations. Phys. Lett. 50 A (1974), 42-44.
[2] Makhankov V.G.: Dynamics of classical solitons (in non-integrable systems). Physics Reports, A review section of Physics Letters (section C), \bf 35(1) (1978), 1-128. MR 0481361
[3] Nishikawa K., Hajo H., Mima K., Ikezi H.: Coupled nonlinear electron-plasma and ion- acoustic waves. Phys. Rev. Lett. 33 (1974), 148-150.
[4] Makhankov V.G.: Preprint. JINR E5-8359, Dubna, 1974.
[5] Bogoluosksy J.L., Makhankov V.G.: Preprint. JINR E4-9425, Dubna, 1975.
[6] Reed M., Simon B.: Method of Modern Mathematical Physics. Academic Press, New York and London, 1972. MR 0493419
[7] Vejvoda O.: Partial Differential Equations: Time-Periodic Solutions. Martinus Nijhoff Publishers, The Hague, Boston, London, 1982. Zbl 0501.35001
[8] Maz'ja V.G.: Sobolev Spaces. Springer-Verlag, 1985. MR 0817985 | Zbl 0692.46023
[9] Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., 1964. MR 0181836 | Zbl 0173.12701
[10] Chen Guowang, Yang Zhijian, Zhao Zhancai: Initial value problems and first boundary problems for a class of quasilinear wave equations. Acta Mathematicae Applicatae Sinica 9 (1993), 289-301. Zbl 0822.35094
[11] Sun Hesheng: On the mixed initial boundary value problem for semilinear degenerate evolution equation (in Chinese). Chinese Annals of Mathematics 8 (A) (1987), 32-43. MR 0901636
[12] Yosida K.: Functional Analysis. Springer, 1978. MR 0500055 | Zbl 0830.46001

Partner of