In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{cc}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)\hfill & 0\<x\<\pi ,t\in ℝ,i\in \mathbb{Z}\left(TC\right)\hfill \\ {\varphi }^i(0,t)={\varphi }^i(\pi ,t)=0\hfill & \forall t,i.\hfill \end{array}\right.$$We consider the case of “closed chains” i.e. ${\varphi }^{i+N}={\varphi }^i\forall i\in \mathbb{Z}$ and some $N\in \mathbb{N}$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{c}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)0<x<\pi ,t\in \mathrm{I}\mathrm{R},\mathrm{i}\in \mathrm{Z}\mathrm{Z}\left(\mathrm{TC}\right){\varphi }^{\mathrm{i}}(0,\mathrm{t})={\varphi }^{\mathrm{i}}(\pi ,\mathrm{t})=0\forall \mathrm{t},\mathrm{i}.\hfill \end{array}\right.$$ We consider the case of “closed chains" i.e.${\varphi }^{i+N}={\varphi }^i\forall i\in ZZ$ and some $N\in IN$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N-wave equation are reviewed. New results on explicit construction of the wave functions for radial...

We give explicit formulas for Hadamard's coefficients in terms of the tau-function of the Korteweg-de Vries hierarchy. We show that some of the basic properties of these coefficients can be easily derived from these formulas.

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $${\widehat{H}}_s^r\left(ℝ\right)$$ defined by the norm $${∥v_0∥}_{{\widehat{H}}_s^r\left(ℝ\right)}:={∥{〈\xi 〉}^s\widehat{v_0}∥}_{L_{\xi }^{r^{\text{'}}}},〈\xi 〉={\left(1+{\xi }^2\right)}^{\frac{1}{2}},\frac{1}{r}+\frac{1}{r^{\text{'}}}=1$$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $$\frac{2j-1}{2r^{\text{'}}}$$ . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < $$\frac{2j-1}{2r^{\text{'}}}$$ . The results for r = 2 - so far in...