Curve motion inducing modified Korteweg-de Vries systems

Curve flows in Lagrange–Finsler geometry, bi-Hamiltonian structures and solitons

“…Considering a prescribedg ab , we have to integrate two times on y e in order to find any solution for e a a defining a frame structure in the vertical subspace. The next step is to construct the d-metricg αβ = [g ij ,g ab ] of type (17), in our case, with respect to a nonholonomic base elongated by N i j , generated by g ij (x) andg ef =g ab , like in (8) and (9). This defines a constant curvature Riemannian space of dimension n + n. The coefficients of the canonical d-connection, which in this case coincide with those for the Levi Civita connection, and the coefficients of the Riemannian curvature can be computed respectively by puttingg ef =g ab in formulas (70) and (75), see Appendix.…”

“…For a regular Lagrangian L(x, y) and corresponding Euler-Lagrange equations it is possible to construct canonically a generalized Finsler geometry on 9 In another direction, there is a proof [32] that any Lagrange fundamental function L can be modeled as a singular case in a certain class of Finsler geometries of extra dimension. Nevertheless the concept of Lagrangian is a very important geometrical and physical one and we shall distinguish the cases when we model a Lagrange or a Finsler geometry: A physical or mechanical model with a Lagrangian is not only a "singular" case for a Finsler geometry but reflects a proper set of geometric objects and structures with possible new concepts in physical theories.…”

“…In parallel, the differential geometry of plane and space curves has received considerable attention in the theory of integrable nonlinear partial differential equations and applications to modern physics [6,7,8,9,10,11,12]. More particularly, it is well known that both the modified Korteweg-de Vries (mKdV) equation and the sine-Gordon (SG) equation can be encoded as flows of the curvature invariant of plane curves in Euclidean plane geometry.…”

Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics

“…10 An N-connection structure (in particular, a fractional Lagrangian) induces an N-anholonomic Klein space (stated by two left-invariant g-and g-valued Maurer-Cartan forms on the Lie d-group G = ( G G)) is identified with the zero-curvature canonical d-connection 1-form …”

“…It is well known that both the modified Korteweg -de Vries equation, (mKdV), and the sine-Gordon, (SG , (solitonic) equations can be encoded as flows of the invariant curvatures invariant of plane curves in Euclidean plane geometry [7,8,[10][11][12]14]. Such constructions were developed [1,3,15] for curve flows in Riemannian manifolds of constant curvature [9,16].…”