The present article reviews methods from spectral theory and algebraic geometry for finding explicit solutions of the Toda equation, namely for the N -soliton solution and quasi-periodic solutions. Along they way basic concepts like Lax pairs, associated hierarchies, and Bäcklund transformations for the Toda equation are introduced.

Using Painleve singularity structure analysis, we show that coupled higher-order nonlinear Schrodinger (CHNLS) equations admit the Painleve property. Using the results of the Painleve analysis, we succeed in Hirota bilinearizing the CHNLS equations for the integrable cases. Solving the Hirota bilinear equations, one soliton and two soliton solutions are explicitly obtained. Lax pairs are explic...

By using the first and second flows of Kowalevski top, we can make top into two which has time variales. Then show that equations become those full genus Jacobi inversion problem. In addition to Lax pair for flow, costruct flow. Using flows, Lie group structure these pairs is Sp(4,$\mathbb{R}$) $\cong$ SO(3,2). Through conclude hyperelliptic function

The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota–Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.

The equations that define the Lax pairs for generalized principal chiral models can be solved for any constant nondegenerate bilinear form on su(2). The solution is dependent on one free variable that can serve as the spectral parameter. Necessary conditions for the nonconstant metric on SU(2) that define the integrable models are given. 1991 MSC numbers: 35L10, 35L15, 34A55

We discuss the non–autonomous nonlinear partial difference equations belonging to Boll classi?cation of quad graph consistent around cube. show how starting from compatible on a cell we can construct lattice equations, its Backlund transformations and Lax pairs. By carrying out algebraic entropy calculations that H4 trapezoidal H6 families are linearizable in few examples effectively linearize ...

We study Padé interpolation problems on an additive grid, related to difference (d-) Painlevé equations of type $$E_7^{(1)}$$ , $$E_6^{(1)}$$ $$D_4^{(1)}$$ and $$A_3^{(1)}$$ . By choosing suitable problems, we can derive time evolution equations, scalar Lax pairs contiguous determinant formulae special solutions given in terms hypergeometric functions, for the corresponding d-Painlevé equations.

The aim of this study was to investigate the relationship between compensatory shortening and coarticulation in German tense and lax vowels in trochees and iambs and to determine whether this relationship was influenced by prosodic accentuation. Speakers produced near minimal pairs differing in vowel tensity in monosyllabic and disyllabic words (both trochees and iambs) in accented and deaccent...