Cross-constrained Variational Problem Research Articles

Small solitons and multisolitons in the generalized Davey-Stewartson system

Abstract By introducing and solving a new cross-constrained variational problem, a one-to-one correspondence from the prescribed mass to frequency of soliton is established for the generalized Davey-Stewartson system in two-dimensional space. Orbital stability of small soiltons depending on frequencies is proved. Multisolitons with different speeds are constructed by stable small solitons.

Global Existence and Blowup for the Cubic Nonlinear Klein-Gordon Equations in Three Space Dimensions

Abstract In this paper, we apply a cross-constrained variational method to study the classic nonlinear Klein-Gordon equation with cubic nonlinearity in three space dimensions. By constructing a type of cross-constrained variational problem and establishing the so-called cross invariant manifolds, we obtain a sharp threshold for blowup and global existence of the solution to the equation under study which is different from that in [10] . On the other hand, we give an answer to the question that how small the initial data have to be for the global solutions to exist.

Sharp criterion of global existence for nonlinear Schrödinger equation with a harmonic potential

This paper discusses nonlinear Schrodinger equation with a harmonic potential. By constructing a different cross-constrained variational problem and the so-called invariant sets, we derive a new threshold for blow-up and global existence of solutions.

Sharp threshold of global existence for the Klein-Gordon equations with critical nonlinearity

In this paper, we present a cross-constrained variational method to study the Cauchy problem of the nonlinear Klein-Gordon equations with critical nonlinearity in two space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish a sharp threshold of global existence and blowup of it. Furthermore, we answer the question: How small are the initial data if the solution exists globally.

A cross-constrained variational problem for the generalized Davey–Stewartson system

AbstractWe study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson systemin ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential

In this paper, we apply a cross-constrained variational approach forthe nonlinear Klein-Gordon equations with an inverse squarepotential in three space dimensions (which is a representative ofthe class of equations of interest) based on the relationshipbetween a type of cross-constrained variational problem and energy.By constructing a type of cross-constrained variational problem andestablishing so-called cross-invariant manifolds of the evolutionflow, we first derive a sharp threshold for global existence andblow-up of solutions to the Cauchy problem for the equations understudy. On the other hand, we get an answer of the question: howsmall are the initial data, the global solutions exist?

CROSS-CONSTRAINED VARIATIONAL PROBLEM AND THE NON-LINEAR KLEIN–GORDON EQUATIONS

AbstractIn this paper, we put forward a cross-constrained variational method to study the non-linear Klein–Gordon equations with an inverse square potential in three space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish some new types of invariant sets for the equation and derive a sharp threshold of blowup and global existence for its solution. Finally, we give an answer to the question how small the initial data are for the global solution to exist.

Sharp conditions of global existence for the quasilinear Schrödinger equation

The paper discusses a class of quasilinear Schrodinger equations describing the upper-hybrid oscillation propagation. By establishing a cross-constrained variational problem and a so-called invariant flow, we obtain a sharp condition for blow-up and global existence of solutions of the Cauchy problem.

Global solution for coupled nonlinear Klein-Gordon system

The global solution for a coupled nonlinear Klein-Gordon system in two-dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.

Sharp conditions of global existence for the generalized Davey–Stewartson system

This paper discusses the generalized Davey–Stewartson system. By constructing a cross-constrained variational problem and so-called invariant manifolds of the evolution flow, we derive a sharp criterion for blow-up and global existence of the solutions.

Nonlinear Schrödinger equation with harmonic potential

This paper discusses a class of nonlinear Schrödinger equation with harmonic potential. By constructing a cross-constrained variational problem and so-called invariant manifolds of the evolution flow, we derive a sharp criterion for blow-up and global existence of the solutions.

Sharp Threshold for Blowup and Global Existence in Nonlinear Schrödinger Equations Under a Harmonic Potential

ABSTRACT We present a variational approach to study the nonlinear Schrödinger equations under a harmonic potential. By constructing a type of cross constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp threshold for global existence and blowup of the solution. The stability of the standing waves is also discussed.