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The inverse spectral method, nonlinear Fourier transforms and integrability of the high-dimensional Date–Jimbo–Kashiwara–Miwa equation

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  • Gui, Linlin
  • Zhang, Yufeng

Abstract

The paper is organized in three parts: (a) We construct the Lax pair of the matrix form of the 2＋1-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation, so that a nonlinear Fourier transform of the Cauchy solution u is obtained, denoted by H. The associated time evolution of u is derived by the time evolution of the nonlinear Fourier data. (b) The complexification of the independent variables x, y, t of the 2＋1-dimensional DJKM equation generate the 4＋2 integrable extension of the DJKM equation, we derive a nonlinear Fourier transform pair in four dimensions, which can be used for the solution of the Cauchy initial value problem of the DJKM equation in 4＋2. (c) Reducing the equation from 4＋2 to the 3＋1 and 3＋2 dimensions by transforming two variables, and the Lax pairs of the reduced equations are given. Finally, the three dimensional Fourier transform pair and solution of the Cauchy problem for the DJKM equation in three spatial and two temporal dimensions is constructed by introducing several new long derivative operators Dx, Dy, and Dt.

Suggested Citation

  • Gui, Linlin & Zhang, Yufeng, 2025. "The inverse spectral method, nonlinear Fourier transforms and integrability of the high-dimensional Date–Jimbo–Kashiwara–Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 199(P3).
    • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p3:s0960077925008975
    DOI: 10.1016/j.chaos.2025.116884
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