¹Department of Mathematics and Physics, ShiJiaZhuang TieDao University, ShiJiaZhuang, China
²Qinggong College, Hebei United University, TangShan, China
³College of Mathematics, Inner Mongolia University for Nationalities, TongLiao, China

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Abstract—Soliton equations are infinite-dimensional integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange function and Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the Bargmann constrained condition between the potential function and the eigenfunction, the Lax pairs is equivalent to matrix spectral problem. Furthermore, the involutive representations of the solutions for long wave equation are generated.

Index Terms—Spectral problem, Hamilton canonical system, Bargmann constraint, integrable system, involutive solution.

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Cite: Wei Liu, Shujuan Yuan, Shuhong Wang, "Nonlinear Evolution Equations for Second-order Spectral Problem," Journal of Computers vol. 7, no. 9, pp. 2144-2151, 2012.