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M. Klaus, D.E. Pelinovsky and V.M. Rothos

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Evans function for Lax operators
with algebraically decaying potentials

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Journal of Nonlinear Science 16, 1-44 (2006)

**Abstract:**

We study the instability of algebraic solitons for integrable
nonlinear equations in one spatial dimension that include modified
KdV, focusing NLS, derivative NLS, and massive Thirring equations.
We develop the analysis of the Evans function that define
eigenvalues in the corresponding Lax operators with algebraically
decaying potentials. The standard Evans function generically has
singularities in the essential spectrum, which may include embedded
eigenvalues with algebraically decaying eigenfunctions. We construct
a renormalized Evans function and study bifurcations of embedded
eigenvalues, when an algebraically decaying potential is perturbed
by a generic potential with a faster decay at infinity. We show that
the bifurcation problem for embedded eigenvalues can be reduced to
cubic or quadratic equations, depending on whether the algebraic
potential decays to zero or approaches a non-zero constant. Roots of
the bifurcation equations define eigenvalues which correspond to
nonlinear waves that are formed from unstable algebraic solitons.

Our results provide precise information on the transformation of
unstable algebraic solitons in the time-evolution problem associated
with the integrable nonlinear equation. Algebraic solitons of the
modified KdV equation are shown to transform to either travelling
solitons or time-periodic breathers, depending on the sign of the
perturbation. Algebraic solitons of the derivative NLS and massive
Thirring equations are shown to transform to travelling and rotating
solitons for either sign of the perturbation. Finally, algebraic
homoclinic orbits of the focusing NLS equation are destroyed by the
perturbation and evolve into time-periodic space-decaying solutions.

**Keywords:**

EVANS FUNCTION, LAX OPERATORS, EMBEDDED EIGENVALUES,
ESSENTIAL SPECTRUM OF LINEAR OPERATORS,
ALGEBRAIC SOLITONS, HOMOCLINIC ORBITS,
MODIFIED KORTEWEG-DE VRIES EQUATION,
FOCUSING NONLINEAR SCHRODINGER EQUATION,
DERIVATIVE NONLINEAR SCHRODINGER EQUATION,
MASSIVE THIRRING MODEL.