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Discrete nonlinear Schr\"odinger type equations: Solutions and continuum limits
Song-lin Zhao, Xiao-hui Feng, Wei Feng
https://arxiv.org/abs/2404.14060 https://arxiv.org/pdf/2404.14060
arXiv:2404.14060v1 Announce Type: new
Abstract: As local and nonlocal reductions of a discrete second-order Ablowitz-Kaup-Newell-Segur equation, two discrete nonlinear Schr\"odinger type equations are considered. Through the bilinearization reduction method, we construct double Casoratian solutions of the reduced discrete nonlinear Schr\"odinger type equations, including soliton solutions and Jordan-block solutions.Dynamics of the obtained one-soliton and two-soliton solutions are analyzed and illustrated. Moreover,both semi-continuous limit and full continuous limit, are applied to obtain solutions of the local and nonlocal semi-discrete nonlinear Schr\"odinger type equations, as well as the local and nonlocal continuous nonlinear Schr\"odinger type equations.

Solutions of local and nonlocal discrete complex modified Korteweg-de Vries equations and continuum limits
Ya-Nan Hu, Shou-Feng Shen, Song-lin Zhao
https://arxiv.org/abs/2404.14150

Solutions of local and nonlocal discrete complex modified Korteweg-de Vries equations and continuum limits
Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations is reconsidered, where two `proper' discrete Ablowitz-Kaup-Newell-Segur equations and two `unproper' discrete Ablowitz-Kaup-Newell-Segur equations are derived. The `proper' equations admit local reduction, while the `unproper' equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two local and nonlocal discrete complex mo…

Riemann--Hilbert method to the Ablowitz--Ladik equation: higher-order case
Huan Liu, Jing Shen, Xianguo Geng
https://arxiv.org/abs/2402.08352 https://

Riemann--Hilbert method to the Ablowitz--Ladik equation: higher-order case
We focused on the Ablowitz--Ladik equation on a zero background, specifically considering the scenario of $N$ pairs of multiple poles. Our first goal was to establish a mapping between the initial data and the scattering data. This allowed us to introduce a direct problem by analyzing the discrete spectrum associated with $N$ pairs of higher-order zeros. Next, we constructed another mapping from the scattering data to a $2\times2$ matrix Riemann--Hilbert problem equipped with several residue co…

Algebraic Complete Integrability of the (a_4^{\left(2\right)}) Toda lattice
B. L. Lietap Ndi (University of Maroua, Faculty of Sciences, Department of Mathematics and Computer Sciences, Cameroon), Djagwa Dehainsala (University of NDjamena, Faculty of Exact and Applied Sciences, Department of Mathematics, Chad), Joseph DONGHO (University of Maroua, Faculty of Sciences, Department of Mathematics and Computer Sciences, Cameroon)

Algebraic Complete Integrability of the (a_4^{\left(2\right)}) Toda lattice
The aim of this work is focused on the investigation of the algebraic complete integrability (a.c.i) of the Toda lattice associated with the twisted affine Lie algebra (a_4^{\left(2\right)}). First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description o…

Algebraic Complete Integrability of the (a_4^{\left(2\right)}) Toda lattice
B. L. Lietap Ndi (University of Maroua, Faculty of Sciences, Department of Mathematics and Computer Sciences, Cameroon), Djagwa Dehainsala (University of NDjamena, Faculty of Exact and Applied Sciences, Department of Mathematics, Chad), Joseph DONGHO (University of Maroua, Faculty of Sciences, Department of Mathematics and Computer Sciences, Cameroon)

Algebraic Complete Integrability of the (a_4^{\left(2\right)}) Toda lattice
The aim of this work is focused on the investigation of the algebraic complete integrability (a.c.i) of the Toda lattice associated with the twisted affine Lie algebra (a_4^{\left(2\right)}). First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description o…