A Class Of Schrodinger Equations And Applications

Schro¨dinger equation is a fundamental partial diferential equation in quantummechanics. The nonlinear Schr¨odinger equation is used in Bose-Einstein condensatetheory. It’s also applied in optics and water waves[60,61,21]. In the present thesis,we study a class of nonlinear Scho¨dinger equation with applications.In Chapter2, we consider some coupled nonlinear Schr¨odinger lattice system. Thissystem is one of the famous models in nonlinear optics. It describes a model of latticesystems of coupled anharmonic oscillators.In Section2.1, we study the excitation threshold of the coupled nonlinear Schr¨odin-ger lattice system:i uË™l+k(du)l+|ul|2ul+|vl|2ul=0,i vË™2l+k(dv)l+|vl|vl+|ul|2vl=0, l∈Zd.wheredis discrete Laplacian operator.Flach et al.[53] are the first who numerically study the excitation thresholds forthe breathers in one-, two-, and three-dimensional lattice. It suggests that for a class ofHamiltonian dynamical systems, there is a lower bound to the power of the soliton if thedimension is greater or equals than a critical value. For discrete nonlinear Schr¨odingerequation, Weinstein[35] discusses a connection among the dimensionality, the degreeof the nonlinearity and the existence of the excitation threshold. If the total power isgreater than the excitation threshold, then the ground state exists. If the total power isless than the excitation threshold, then the ground state doesn’t exist. The excitationthreshold of the lattice system is studied by the variational method. We apply theconcentration compactness principle in the coupled case. Our aim is to find the relationship between the existence of the excitation thresholdand the dimensionality d of space for the coupled lattice system. It’s proved that ifd=1, then the excitation threshold doesn’t exist; if d≥2, then the excitation thresholdexists. We also establish a upper bound of the excitation threshold.In Section2.2, we consider the existence of the soliton to the coupled nonlinearSchr¨odinger lattice system with saturable nonlinearity:2i uË™u2l(|ul|+|vl|)l+(du)l+β1+|u,l|2+|v=l|20i vË™|2+|v2l|)l+(dv)l(|ull+βv1+|ul|2+|vl|2=0,|l|≤K,where β>0. Due to the lack of compactness for the saturable nonlinearity, we considerthe following boundary condition:ul=u l, vl=v l,|l|≤K, and uK+1=u (K+1)=vK+1=v (K+1)=0.Our aim is to obtain the soliton by Lagrange method. The existence of the standingwave with prescribed frequencies is proved by Nehari manifold method. Moreover, thelower total power of the standing wave with prescribed frequencies is estimated bycontracting mapping theorem and Lax-Milgram theorem.In Chapter3, we consider a class of fractional nonlinear Sch¨odinger equation.The fractional Schr¨odinger equation is a fundamental equation of fractional quantummechanics. It is formulated by Nick Laskin as a result of expanding the Feynmanpath integral, from the Brownian-like to the L′evy-like quantum mechanical paths.It suggests that a path integral over the L′evy-like quantum-mechanical paths giverise to a generalized quantum mechanics. If the Feynman path integral leads to theclassical Schr¨odinger equation, then the path integral over L′evy trajectories leads tothe fractional Schr¨odinger equation. In the last decades, a lot of attention has beenpaid to study the fractional nonlinear Schr¨odinger equation.In Section3.1, we consider the fractional nonlinear Schr¨odinger equation withunbounded potentiali tψ=()αψ+V (x)ψ|ψ|p1ψ, x∈Rd, t∈R, where0<α <1, ψ is the wave function, V (x) is the unbounded potential. We knowthat if α=1, then the fractional Schro¨dinger equation is the fundamental Schr¨odingerequation.Patricio Felmer et al.[46] considered the positive solutions of nonlinear fractionalSchro¨dinger equation without potential. They prove the regularity, decay and sym-metry properties of the standing wave solutions. Zhang[14] study the ground state ofclassical nonlinear Schr¨odinger with unbounded potential.Our work is to prove the existence of ground state for the nonlinear Schr¨odingerequation with unbounded potential by Lagrange multiplier method. And we use Neharimanifold argument to obtain the soliton with prescribed frequency. We also prove thatthe soliton is bound state, i.e.,|ψ|â†'0, as|x|â†'∞.In Section3.2, we concern the stability of the ground state for the following non-linear Schr¨odinger equation:idudt=()αu|u|p1u, x∈R,where0<α <1.The stability of ground state for classical Schr¨odinger equation and Korteweg-deVries equation is discussed in [17,59,37,38,27,32,33,34]. This result is obtained bycompactness method or setting Lyapunov functional.Our aim is to study the stability of ground state for the fractional nonlinearSchro¨dinger equation in one dimension. First, we show global well-posedness of theCauchy problem of the fractional nonlinear Schr¨odinger equation. Then, we present theexistence of the ground state by the variational method. At last, we use the Lyapunovfunctional to obtain the orbital stability of the ground state.In Section3.3, we study the long time behavior of solutions for the coupled dissi-pative fractional Schr¨odinger equation:i uit=()αui iδui mj=1βij|u1qi|q ui|uj|+1+fi(x), x∈R, t∈R+,ui|t=0=ui(0)∈Hα(R), where δ>0,12<α <1, βij=βji, i=1,..., m. And fi(x)∈Hα(R)=0is the drivingterm.We prove that if βij<0,1≤q <2α, there exists a global attractor. A technicalkey to obtain the attractor is the existence of bounded absorbing set and the asymptoticcompactness. Since the solutions of system on the complement of a sufciently largeball are uniformly small for large time. This together with the energy equations yieldsthe asymptotic compactness of the solutions. And we use a priori estimate of thesolutions to obtain the existence of the bounded absorbing set.