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Do symmetry constraints yield exact solutions? (English) Zbl 1137.35325
In the present study the authors would like to show, through a study of symmetry constraints of the Sharma-Tassa-Olver equation, that not all symmetry constraints of differential equations yield exact solutions of the equations. They explain why such phenomenon can happen in the symmetry theory so that mistakes can be avoided in constructing exact solutions by the symmetry constrained method.

35C05Solutions of PDE in closed form
35A30Geometric theory for PDE, characteristics, transformations
35Q53KdV-like (Korteweg-de Vries) equations
58J70Invariance and symmetry properties
Full Text: DOI
[1] Ma, W. X.; Strampp, W.: An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems. Phys lett A 185, 277-286 (1994) · Zbl 0941.37530
[2] Sharma AS, Tasso H. Report IPP 6/158 Ber. MPI Plasmaphysik, Garching. p. 1-10.
[3] Olver, P. J.: Evolution equations possessing infinitely many symmetries. J math phys 18, 1212-1215 (1977) · Zbl 0348.35024
[4] Lian, Z. J.; Lou, S. Y.: Symmetries and exact solutions of the Sharma-Tasso-Olver equation. Nonlinear anal 63, No. 5-7, e1167-e1177 (2005)
[5] Yang, Z. J.: Travelling wave solutions to nonlinear evolution and wave equations. J phys A: math gen 27, 2837-2855 (1994) · Zbl 0837.35034
[6] Gudkov, V. V.: A family of exact travelling wave solutions to nonlinear evolution and wave equations. J math phys 38, 4794-4803 (1997) · Zbl 0886.35131
[7] Ma, W. X.; Zhou, Z. X.: Binary symmetry constraints of N-wave interaction equations in 1+1 and 2+1 dimensions. J math phys 42, 4345-4382 (2001) · Zbl 1063.37065
[8] Ma, W. X.; Zhou, R. G.: Adjoint symmetry constraints leading to binary nonlinearization. J nonlinear math phys 9, No. Suppl. 1, 106-126 (2002)