Solutions of Non-Linear Differential Equation (<U>Remark: Graphics omitted.</U>)

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  • Yamamoto Yosinori
    Institute of Precision Mechanics, Faculty of Engineering, Hokkaido University
  • Takizawa Éi Iti
    Institute of Precision Mechanics, Faculty of Engineering, Hokkaido University

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  • Solutions of Non-Linear Differential Equation \(\dfrac{\partial u}{\partial t}+\dfrac{45}{2} \delta ^{2}u^{2} \dfrac{\partial u}{\partial x}-\dfrac{\partial ^{5}u}{\partial x^{5}}=0\)

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The non-linear partial differential equation:<BR>ut+(45⁄2)δ2u2 ux−uxxxxx=0,<BR>has solutions expressed by elliptic functions. A solitary wave solution is also found to be u=±\sqrtκ⁄3δ[3 \sech2 (A(ξ−ξ0))−1], with ξ=x−(9δκt⁄2), A=\sqrtδ(e1−e3)⁄2, e1=−2e3=2\sqrtκ⁄(3δ), and any constants ξ0 and κ.

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