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Conservation laws for a generalized seventh order KdV equation. (English) Zbl 1433.35331

Using the multiplier method, the authors obtain a complete classification of local lower order conservation laws for a generalized third order Korteweg-de Vries equation (KdV) based on seven non-zero arbitrary parameters; more precisely, \[u_t+a_1u^3u_x+a_2u_x^3+a_3uu_xu_{2x}+a_4u^2u_{3x}+a_5u_{2x}u_{3x}\] \[+a_6u_xu_{4x}+a_7uu_{5x}+u_{7x}=0,\] where \(a_1,\dots,a_7\) are non zero parameters and \(u_{2x}=\frac{\partial^2u}{\partial x^2}\),…,\(u_{7x}=\frac{\partial^7u}{\partial x^7}\). The authors determine the parameters \(a_1,\dots,a_7\) for which the KdV equation admits multipliers and for each multiplier, they construct the conservation laws of this equation. They apply the Lie method in order to classify all point symmetries admitted by the equation in terms of the arbitrary parameters and find that there are no special cases of the parameters for which the equation admits extra symmetries, other than those that can be found by inspection (scaling symmetry and space and time translation symmetries). They consider the reduced ordinary differential equations and they determined all the integration factors of the reduced equation from the combined \(x\)- and \(t\)-translation symmetries. They observe that all integrating factors arise by reduction of the low-order multipliers of the generalized seventh-order KdV equation. Finally, they find the integrating factors of the reduced ODE.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
22E30 Analysis on real and complex Lie groups
34C14 Symmetries, invariants of ordinary differential equations
58J20 Index theory and related fixed-point theorems on manifolds

Software:

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References:

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