Geometrical Construction of the Hirota Bilinear Form of the Modified Korteweg-De Vries Equation on a Thin Elastic Rod: Bosonic Classical Theory

Abstract

Recently there have been several studies of a nonrelativistic elastic rod in R2 whose dynamics is governed by the modified Korteweg-de Vries (MKdV) equation. Goldstein and Petrich found the MKdV hierarchy through its dynamics [Phys. Rev. Lett. 69, 555 (1992).] In this article, we will show the physical meaning of the Hirota bilinear form along the lines of the elastica problem after we formally complexify its arc length.

IMMERSION ANOMALY OF DIRAC OPERATOR ON SURFACE IN ℝ3

“…Recently Goldstein and Petrich discovered the modified KdV (MKdV) hierarchy by considering one parameter deformation of a space curve immersed in R 2 [12,13]. After their new interpretation, there appear several geometrical realizations of the soliton theory [14][15][16]. In the differential geometry, after the discovery the exotic solution of the constant mean curvature surface by Wente [17], the extrinsic structure is currently studied again [18][19][20].…”

“…On the other hand I have been studying the Dirac field confined in an immersed object and its relationship with the immersed object itself [28][29][30][31][32]. Since the Dirac operator should be regarded as a translator (a functor) between the analytical object and the geometrical object [8][9], in terms of the Dirac operator, I have been studying the physical and geometrical meanings of the abstract theorems in the soliton theory and quantum theory focusing on the elastica problem [28][29][30][31][32] and recently on the immersed surface [1].…”

“…Here I will evaluate the variations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) in the framework of the quantum theory [30,31,33],…”

“…From (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18), the Ward-Takahashi identity [29,30],…”

“…From the soliton theory, the MNV equation is a higher dimensional analogue of the MKdV equation [6]. The Dirac operator appearing in the auxiliary linear problems of the MKdV equation is realized as the operator for the Dirac field confined in the elastica [28][29][30][31][32] as the KKWE equation might be related to auxiliary linear problems of the MNV equation [2,6] and is realized as the equation of the Dirac field confined in the immersed surface [1].…”