Stability of Solutions to the Discrete Nonlinear Schrodinger Equation in Multiple Dimensions

1
Stability of Solutions to the Discrete Nonlinear
Schrodinger Equation in Multiple Dimensions

• Jacob A Gagnon
• PhD Student
• UMASS Amherst Mathematics

2
Overview of Talk

• How to find time periodic numerical solutions in
  one dimension
• Time periodic numerical solutions in two and
  three dimensions
• Stability of these solutions
• Regions of Stability vs Instability in Parameter
  Space
• Future Work

3
One dimension

• 1-D DNLS

• u is a complex valued function, c is inverse of
  the square of the lattice spacing, n is nodal
  index
• For time periodic solutions Use

• Can be solved for c 0 (anti continuum limit)
  for each n as

• Use the c 0 solution to solve the nonzero c
  problem

4
One dimension continued

• We expect the c 0.001 solution to be nearby the
  c 0 solution
• Notice that the above equation is set of
  simultaneous nonlinear equations, one equation
  for each nodal location
• These can be solved through Newton iteration
• Apply the Dirchlet Boundary condition
• Use the c 0 solution as an initial guess for
  Newton for the c 0.001 solution
• After Newton obtains the c0.001 solution, use
  this solution as an initial guess for the c0.002
  solution
• Continue this process until the desired c value
  is reached

5
One Dimensional results

• C 0.15, Lambda 1
• Three differential initial conditions
• UR ,
• LL , -
• LR , , -

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Stability of 1-D solutions

• Study the behavior of perturbations to a time
  periodic solution
• 0th order
• 1st order
• Try the form
• Eigenvalue problem

7
Stability Results

• C 0.15, Lambda 1
• Three differential initial conditions
• UR ,
• LL , -
• LR , , -

8
Two dimensions and more

• 2-D DNLS has the form
• Again we search for time periodic solutions of
  the form
• u is complex valued so decompose into real and
  imaginary parts

9
Two dimensions continued

• Again use c 0 analytic solution for initial
  guess in Newton iteration to find c 0.001
  solution. Newton takes the form
• V is the vector
• J is the Jacobian of the system, j iteration
  index, F is the nonlinear system

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Evaluation of Jacobian and F

• Care must be taken in evaluation of Jacobian and
  F vector due to Dirchlet BC
• We must pay attention to nodal location
• Is the node a central node, edge node, or corner
  node?
• Ex. on the bottom edge , 0
• In the Jacobian, for central nodes nonzero
  matrix entries are
• I is the indexing function that assigns each
  nodal location to an index in the v vector
• is (j - 1) n i and is (j 1) n i
  n n
• For edge or corner nodes some of the above matrix
  entries are zero

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2-D Results Part I

• 1st example
• Vortex initial condition
• -i -1 i
• 1 0 1 i -1 -i
• 20 by 20 grid, lambda 1.0, c .100
• Re, Im, Magnitude, and Phase plots

12
2D Results Part II

• 2nd example
• Vortex initial condition
• -1 i
• -i 1
• 20 by 20 grid, lambda 1.0, c .600
• Real, Im, Magnitude, and Phase plots

13
Stability in 2-D

• Study the stability through an expansion
• 0th order
• 1st order
• Decompose as

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Stability Results in 2D
Re versus c
Fixed c
Im versus c

• Case 1
• C 0.100
• Vortex initial condition
• -i -1 i
• 1 0 1
• i -1 -i

Fixed c
Re versus c
Im versus c

• Case 2
• C 0.600
• Vortex initial condition
• -1 i
• -i 1

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Solutions in Three dimensions

• The 3-D NLS has the form
• Look for time periodic solutions
• Decompose into real and imaginary parts
• Again use multiple rounds of Newton iteration to
  solve, form Jacobian, apply BC (many cases)

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Stability in 3D

• DNLS 3D has the form
• Expansion of the form
• 0th order
• 1st order
• Decompose as
• Eigenvalue problem

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Stability Results in 3D

• Study the stability of the time periodic 3-D
  solutions when the initial condition is root
  lambda at a single site and 0 elsewhere
• Under these conditions with 101010 grid and
  Lambda 0.050 we have
• Bifurcation at 0.050

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Stability in 3D part 2

• Repeat above for many lambda
• Same conditions 3D time periodic, single site
  root lambda,10 by 10 by 10 grid
• Allows us to find boundary between stability and
  instability in Lambda, c parameter space
• Above the line is unstable, Below the line is
  stable

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In 2D, we have

• Similar bifurcation observed in 2D
• 20 by 20 grid, same initial condition

20
Future Directions

• Study time dependence of non time periodic
  solutions
• Try to prove analytically the boundary between
  stability and instability
• Compare these numerical results with other
  numerical schemes

21
Acknowledgments, Citations

• My advisor Professor Kevrekidis
• M. Johansson and S. Aubry, Growth and decay of
  nonlinear Schrodinger breathers interacting with
  internal modes or standing wave phonons'',
  Physical Review E, Volume 61, Number 5, May 2000,
  pg. 5864
• D.E. Pelinovsky, P.G. Kevrekidis, D.J.
  Frantzeskakis, Nonlinear Schrodinger lattices
  I Stability of discrete solitons'', e-Print
  archive, 7 Dec 2004
• D.E. Pelinovsky, P.G. Kevrekidis, D.J.
  Frantzeskakis, Nonlinear Schrodinger lattices
  II Persistence and stability of discrete
  vortices'', e-Print archive, 6 Nov 2004
• R. Carretero Gonzalez, P.G. Kevredkidis, B.A.
  Malomed, D.J. Frantzeskakis, Three dimensional
  Nonlinear Lattices From Oblique Vortices and
  Octupoles to Discrete Diamonds and Vortex
  Cubes'', Physical Review Letters, Volume 94, 27
  May 2005, pg. 203901
• C. Huepe, L.S. Tuckerman, S. Metens, M.E.
  Brachet, Stability and decay rates of
  nonisotropic attractive Bose-Einstein
  condensates'', Physical Review A, Volume 68,
  2003, pg. 023609
• B.A. Malomed, P.G. Kevrekidis, Discrete vortex
  solitons'', Physical Review E, Volume 64, 2001,
  pg. 026601
• P.G. Kevrekidis, I.G. Kevrekidis, Hamiltonian
  Recursive Projection Method''
• P.G. Kevrekidis, D.J. Frantzeskakis, R.
  Carretero-Gonzalez, B.A. Malomed, A.R. Bishop,
  Discrete Solitons and Vortices on Anisotropic
  Lattices'', Physical Review E, July 2005