Browsing by Subject "Evolution equations"
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On the Classification of Fifth Order Quasi-linear Non-constant Separant Scalar Evolution Equations of the KdV-Type Authors:
Publisher and Date:(Physical Soc Japan, 2012)Fifth order, quasi-linear, non-constant separant evolution equations are of the form u(t) = A(partial derivative(5)u/partial derivative x(5)) + (B) over tilde, where A and (B) over tilde are functions of x, t, u and of the derivatives of u with respect to x up to order 4. We use the existence of a "formal symmetry'', hence the existence of "canonical conservation laws'' rho((i)), i = -1, . . . , 5 as an integrability test. We define an evolution equation to be of the KdV-Type, if all odd numbered ...
Publisher and Date:(IOP Publishing Ltd, 2017)The ` separant' of the evolution equation u(t) = F where F is some differentiable function of the derivatives of u up to order m is the partial derivative partial derivative F/partial derivative u(m) where um u(m) = partial derivative(m)u/partial derivative x(m). As an integrability test we use the formal symmetry method of Mikhailov-Shabat-Sokolov which is based on the existence of a recursion operator as a formal series. The solvability of its coefficients in the class of local functions gives ...