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Register a new userOn a new integrable discretization of the derivative nonlinear Schrodinger (Chen-Lee-Liu) equation
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Takayuki Tsuchida
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We propose a general integrable lattice system involving some free parameters, which contains known integrable lattice systems such as the Ablowitz-Ladik discretization of the nonlinear Schrodinger (NLS) equation as special cases. With a suitable choice of the parameters, it provides a new integrable space-discretization of the derivative NLS equation known as the Chen-Lee-Liu equation. Analogously to the continuous case, the space-discrete Chen-Lee-Liu system possesses a Lax pair and admits a complex conjugation reduction between the two dependent variables. Thus, we obtain a proper space-discretization of the Chen-Lee-Liu equation defined on the three lattice sites $n-1$, $n$, $n+1$ for the first time. Considering a negative flow of the discrete Chen-Lee-Liu hierarchy, we obtain a proper discretization of the massive Thirring model in light-cone coordinates. Multicomponent generalizations of the obtained discrete equations are straightforward because the performed computations are valid for the general case where the dependent variables are vector- or matrix-valued.
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The action of a Backlund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary Backlund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schrodinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because Backlund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a new Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices.
The second-type derivative nonlinear Schrodinger (DNLSII) equation was introduced as an integrable model in 1979. Very recently, the DNLSII equation has been shown by an experiment to be a model of the evolution of optical pulses involving self-steepening without concomitant self-phase-modulation. In this paper the $n$-fold Darboux transformation (DT) $T_n$ of the coupled DNLSII equations is constructed in terms of determinants. Comparing with the usual DT of the soliton equations, this kind of DT is unusual because $T_n$ includes complicated integrals of seed solutions in the process of iteration. By a tedious analysis, these integrals are eliminated in $T_n$ except the integral of the seed solution. Moreover, this $T_n$ is reduced to the DT of the DNLSII equation under a reduction condition. As applications of $T_n$, the explicit expressions of soliton, rational soliton, breather, rogue wave and multi-rogue wave solutions for the DNLSII equation are displayed.
We generate hierarchies of derivative nonlinear Schrodinger-type equations and their nonlocal extensions from Lie algebra splittings and automorphisms. This provides an algebraic explanation of some known reductions and newly established nonlocal reductions in integrable systems.
We consider an integrable generalization of the nonlinear Schrodinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.
An integrable semi-discretization of the Camassa-Holm equation is presented. The keys of its construction are bilinear forms and determinant structure of solutions of the CH equation. Determinant formulas of $N$-soliton solutions of the continuous and semi-discrete Camassa-Holm equations are presented. Based on determinant formulas, we can generate multi-soliton, multi-cuspon and multi-soliton-cuspon solutions. Numerical computations using the integrable semi-discrete Camassa-Holm equation are performed. It is shown that the integrable semi-discrete Camassa-Holm equation gives very accurate numerical results even in the cases of cuspon-cuspon and soliton-cuspon interactions. The numerical computation for an initial value condition, which is not an exact solution, is also presented.
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