On the initial value problem for a class of nonlinear integral evolution equations generated through a Riemann-Hilbert spectral problem

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SOLITONS AND PHONONS AS TRUNCATED ASYMPTOTIC SOLUTIONS OF INTEGRABLE SYSTEMS L. Martinez Alonso, i i Departamento de M~todos Matematicos de la Fisica, Universitad Complutense de Madrid, Madrid, Spain A method for characterizing the asymptotic modes of integrable nonlinear equations is given. Solitons and radiation are described in terms of truncated asymptotic solutions and their interacting properties are analyzed.

ON THE INITIAL VALUE PROBLEM FOR A CLASS OF NONLINEAR INTEGRAL EVOLUTION EQUATIONS GENERATED THROUGH A RIEMANN-HILBERT SPECTRAL PROBLEM. P. Santini, M. J. Ablowitz*, A. S. Fokas* Dipartimento di Fisica, Universita di Roma I, Rome, Italy *Department of Mathematics, Clarkson University, Potsdam, NY13676, USA We present the Inverse Spectral Transform scheme for the following class of matrix nonlinear integral equations

# Qt = (]3f(L)Q' Q = Q(x,t) =(q21(x,t)

q12(x't)o

\

where

LF = i(]3(/I+Q2 HF - ~I QH([Q,F]/ Z +Q2)), F = F(x) off diag.,

I 0), (]3 = ( 0 - I

(Hg)(x) = ~I p

f~-~ dy(y_x)-Ig(y)

The associated spectral problem: (x,z,t) = A(x,z,t) ~+(x,z,t),

x e R,

A(x,z,t) = I + z(]3 + (/1+q12q21 - I)I + Q(x,t) is a matrix homogeneous Riemann-Hilbert problem on the line

Imx=O

of the

complex x plane. The solutions ~±(x,z) of this spectral problem exhibit a new type of singularity structure in z, consisting of polar singularities clustering in finite points of the complex z-plane.