TRACE FORMULAS AND CONSERVATION LAWS FOR NONLINEAR EVOLUTION EQUATIONS

1994 ◽  
Vol 06 (01) ◽  
pp. 51-95 ◽  
Author(s):  
F. GESZTESY ◽  
H. HOLDEN
Keyword(s):  
Conservation Laws ◽  
Long Range ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Schrödinger Operators ◽  
Linear Operators ◽  
Nonlinear Evolution Equations ◽  
Schrodinger Operators ◽  
Trace Formulas ◽  
One Dimensional

New trace formulas for linear operators associated with Lax pairs or zero-curvature representations of completely integrable nonlinear evolution equations and their relation to (polynomial) conservation laws are established. We particularly study the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the sine–Gordon equation, and the infinite Toda lattice though our methods apply to any element of the AKNS–ZS class. In the KdV context, we especially extend the range of validity of the infinite sequence of conservation laws to certain long-range situations in which the underlying one-dimensional Schrödinger operator has infinitely many (negative) eigenvalues accumulating at zero. We also generalize inequalities on moments of the eigenvalues of Schrödinger operators to this long-range setting. Moreover, our contour integration approach naturally leads to higher-order Levinson-type theorems for Schrödinger operators on the line.

scholarly journals Conservation Laws of Nonlinear-Evolution Equations

1974 ◽  
Vol 52 (3) ◽  
pp. 886-889 ◽  
Author(s):  
K. Konno ◽  
H. Sanuki ◽  
Y. H. Ichikawa
Keyword(s):  
Conservation Laws ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations

scholarly journals LOCALIZATION FOR ONE DIMENSIONAL LONG RANGE RANDOM HAMILTONIANS

1999 ◽  
Vol 11 (01) ◽  
pp. 103-135 ◽  
Author(s):  
VOJKAN JAKŠIĆ ◽  
STANISLAV MOLCHANOV
Keyword(s):  
Long Range ◽  
Spectral Properties ◽  
Schrödinger Operators ◽  
Pure Point ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Free Part ◽  
Random Hamiltonians

We study spectral properties of random Schrödinger operators hω=h0+vω(n) on l2(Z) whose free part h0 is long range. We prove that the spectrum of hω is pure point for typical ω whenever the off-diagonal terms of h0 decay as |i-j|-γ for some γ>8.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

2018 ◽  
Vol 32 (02) ◽  
pp. 1850016
Author(s):  
Siqi Xu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Compatibility Condition ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Matrix Spectral Problem

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

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