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August  2017, 37(8): 4239-4247. doi: 10.3934/dcds.2017181

On nonlocal symmetries generated by recursion operators: Second-order evolution equations

M. Euler 1, , N. Euler 1,, and  M. C. Nucci 2,

1. 

Division of Mathematics, Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden
2. 

Dipartimento di Matematica e Informatica, Università di Perugia, 06123, Perugia, Italy

* Corresponding author: norbert@ltu.se

Received  February 2017 Revised  May 2017 Published  April 2017

We introduce a new type of recursion operator suitable to generate a class of nonlocal symmetries for those second-order evolution equations in $1+1$ dimension which allow the complete integration of their time-independent versions. We show that this class of evolution equations is $C$-integrable (linearizable by a point transformation). We also discuss some applications.

Keywords: Nonlocal symmetries, recursion operators, evolution equations.
Mathematics Subject Classification: Primary: 35G20, 35A30, 58J70.
Citation: M. Euler, N. Euler, M. C. Nucci. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4239-4247. doi: 10.3934/dcds.2017181
References:
[1]

S. C. Anco and G. Bluman, Direct construction method for conservation laws of PDEs Part Ⅱ: General treatment, Euro. J. Applied Mathematics, 13 (2002), 567-585.  doi: 10.1017/S0956792501004661.  Google Scholar

[2]

M. Euler and N. Euler, Second-order recursion operators of third-order evolution equations with fourth-order integrating factors, J. Nonlinear Math. Phys., 14 (2007), 313-315.  doi: 10.2991/jnmp.2007.14.3.2.  Google Scholar

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N. Euler and M. Euler, On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: Two linearisable hierarchies, J. Nonlinear Math. Phys., 16 (2009), 489-504.  doi: 10.1142/S1402925109000509.  Google Scholar

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M. Euler, N. Euler and N. Petersson, Linearisable hierarchies of evolution equations in (1+1) dimensions, Stud. Appl. Math., 111 (2003), 315-337.  doi: 10.1111/1467-9590.t01-1-00236.  Google Scholar

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A. S. Fokas, Symmetries and Integrability, Stud. Appl. Math., 77 (1987), 253-299.  doi: 10.1002/sapm1987773253.  Google Scholar

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P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys., 18 (1977), 1212-1215.  doi: 10.1063/1.523393.  Google Scholar

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P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[8]

N. Petersson, N. Euler and M. Euler, Recursion Operators for a Class of Integrable ThirdOrder Evolution Equations, Stud. Appl. Math., 112 (2004), 201-225.  doi: 10.1111/j.0022-2526.2004.01511.x.  Google Scholar

show all references

References:
[1]

S. C. Anco and G. Bluman, Direct construction method for conservation laws of PDEs Part Ⅱ: General treatment, Euro. J. Applied Mathematics, 13 (2002), 567-585.  doi: 10.1017/S0956792501004661.  Google Scholar

[2]

M. Euler and N. Euler, Second-order recursion operators of third-order evolution equations with fourth-order integrating factors, J. Nonlinear Math. Phys., 14 (2007), 313-315.  doi: 10.2991/jnmp.2007.14.3.2.  Google Scholar

[3]

N. Euler and M. Euler, On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: Two linearisable hierarchies, J. Nonlinear Math. Phys., 16 (2009), 489-504.  doi: 10.1142/S1402925109000509.  Google Scholar

[4]

M. Euler, N. Euler and N. Petersson, Linearisable hierarchies of evolution equations in (1+1) dimensions, Stud. Appl. Math., 111 (2003), 315-337.  doi: 10.1111/1467-9590.t01-1-00236.  Google Scholar

[5]

A. S. Fokas, Symmetries and Integrability, Stud. Appl. Math., 77 (1987), 253-299.  doi: 10.1002/sapm1987773253.  Google Scholar

[6]

P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys., 18 (1977), 1212-1215.  doi: 10.1063/1.523393.  Google Scholar

[7]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[8]

N. Petersson, N. Euler and M. Euler, Recursion Operators for a Class of Integrable ThirdOrder Evolution Equations, Stud. Appl. Math., 112 (2004), 201-225.  doi: 10.1111/j.0022-2526.2004.01511.x.  Google Scholar

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