Inverse scattering up to smooth functions for the Dirac-ZS-AKNS system

Abstract.

We consider the Dirac-ZS-AKNS system $\psi_{1x} = -ik\psi_1 + q_1(x)\psi_2,$(1)$\psi_{2x} = -ik\psi_2 + q_2(x)\psi_1,$ where $q_1(x), q_2(x)\in W^{n,1}({\Bbb R})$ (the space of functions with n derivatives in L ¹), $n\in{\Bbb N}\cup 0.$(2) We consider for (1) the transition matrix $T(k)=\biggl({a(k)\atop b(k)} {c(k)\atop d(k)}\biggr), k\in {\Bbb R},$ and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case $q_2(x)=\overline{q_1(x))}$ the scattering matrix $S(k)=\biggl({s_{11}(k) \atop s_{21}(k)} {s_{12}(k)\atop s_{22}(k)}\biggr)= {1\over a(k)} \biggl({1\atop -{c(k)}} {b(k)\atop 1}\biggr),\, k\in {\Bbb R}.$ We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions, of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with $n \in {\Bbb N}$, the following formulas are valid: $q_1(x)=2\check{c}(2x) \quad {\rm in} \quad W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}),$(3)$q_2(x)=2\check{b}(-2x) \quad {\rm in} \quad W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}),$ and, in addition, for the case of the Dirac system $q_1(x)=-2\check{s}_{21}(2x) \quad {\rm in} \quad W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}),$(4)$q_2(x)=2\check{s}_{12}(-2x) \quad {\rm in} \quad W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}),$ where $\check{\varphi}(x)=(2\pi)^{-1}\int_{\Bbb R} e^{-ikx}\varphi(k)dk, \, W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R})$ denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac system under condition (2) with $n\in {\Bbb N}$ III. As applications of the results mentioned above, we show that 1) for any real-valued initial data $\theta(0,x)\in W^{n,1}({\Bbb R}), n\in{\Bbb N}$, the Cauchy problem for the sh-Gordon equation $\theta_{xt}=sh\,\theta$ has a unique solution $\theta(t,x)$ such that $\theta_x(t,x)\in C([0,\infty[, W^{n-1,1}({\Bbb R}))$ and $\theta(t,x)\rightarrow 0 \,{\rm as}\,|x|\rightarrow\infty$ for any t > 0, 2) in addition, for $n\in{\Bbb N}$, for such a solution the following formula is valid: $\theta(t,x)=\theta(0,x)\quad {\rm in} \quad W^{n,1}_{\rm loc} ({\Bbb R})/W^{n+1,1}_{\rm loc}({\Bbb R}),$ where $W^{n,1}_{\rm loc}({\Bbb R})$ denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results.