More General Nonlinear Schr (?) Dinger Equation Finite Difference Method With Five Items

In this paper, we will discuss initial-boundary value problem of a more generalnon-linear Schr¨odinger equation with quintic term.boundary conditionand initial conditionq(s), f(x, t) are known real functions, i2=1, β is the constant that is greaterthan zero. Q(s)=∫s0q(σ)dσ, Q(s)≤Asp+B, s≥0,0≤p <3, A and B arenonnegative constants.Non-linear Schr¨odinger equation plays a very important role in high-energyphysics, quantum mechanics, nonlinear optics, superconductivity and deep waterwaves. What’s more, with its continually expanding application range, peopledeepen their studies.This paper mainly solves the following questions:Firstly, We will raise a type of conservative diference scheme according to theinitial-boundary value problem.Secondly, We need to prove that the diference scheme has two important con-served quantity. We still conduct a prior estimate of L2and H1model by usingthe Sobolev embedding theorem and interpolation inequality, so that we can get L∞model estimate of the diference solution.Finally, We will prove the stability of diference scheme and convergence ofdiference solution.