scholarly journals A sharp bilinear restriction estimate for the sphere and its application to the wave-Schrödinger system

2019 ◽  
Author(s):  
Takafumi Akahori
Keyword(s):  
Restriction Estimate ◽  
Schrödinger System

scholarly journals TRACKING DOWN LOCALIZED MODES IN PT-SYMMETRIC HAMILTONIANS UNDER THE INFLUENCE OF A COMPETING NONLINEARITY

2014 ◽  
Vol 54 (2) ◽  
pp. 79-84 ◽  
Author(s):  
Bijan Bagchi ◽  
Subhrajit Modak ◽  
Prasanta K. Panigrahi
Keyword(s):  
Growth Rate ◽  
Refractive Index ◽  
Stability Analysis ◽  
Linear Stability Analysis ◽  
Time Reversal ◽  
Time Parameter ◽  
Optical Systems ◽  
Localized Modes ◽  
Symmetric Structures ◽  
Schrödinger System

The relevance of parity and time reversal (PT)-symmetric structures in optical systems has been known for some time with the correspondence existing between the Schrödinger equation and the paraxial equation of diffraction, where the time parameter represents the propagating distance and the refractive index acts as the complex potential. In this paper, we systematically analyze a normalized form of the nonlinear Schrödinger system with two new families of PT-symmetric potentials in the presence of competing nonlinearities. We generate a class of localized eigenmodes and carry out a linear stability analysis on the solutions. In particular, we find an interesting feature of bifurcation characterized by the parameter of perturbative growth rate passing through zero, where a transition to imaginary eigenvalues occurs.

Schrödinger systems with quadratic interactions

2019 ◽  
Vol 21 (08) ◽  
pp. 1850077
Author(s):  
Rushun Tian ◽  
Zhi-Qiang Wang ◽  
Leiga Zhao
Keyword(s):  
Variational Formulation ◽  
Variational Problems ◽  
Nontrivial Solutions ◽  
New Techniques ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Coupled Schrödinger System ◽  
Schrödinger Systems ◽  
Bose Einstein ◽  
Schrödinger System

In this paper, we consider the existence and multiplicity of nontrivial solutions to a quadratically coupled Schrödinger system [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are constants and [Formula: see text], [Formula: see text]. Such type of systems stem from applications in nonlinear optics, Bose–Einstein condensates and plasma physics. The existence (and nonexistence), multiplicity and asymptotic behavior of vector solutions of the system are established via variational methods. In particular, for multiplicity results we develop new techniques for treating variational problems with only partial symmetry for which the classical minimax machinery does not apply directly. For the above system, the variational formulation is only of even symmetry with respect to the first component [Formula: see text] but not with respect to [Formula: see text], and we prove that the number of vector solutions tends to infinity as [Formula: see text] tends to infinity.

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