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Superposition formulas for vector generalizations of the mKdV equation. (English. Russian original) Zbl 1156.35456
Math. Notes 82, No. 4, 448-450 (2007); translation from Mat. Zametki 82, No. 4, 501-503 (2007).
From the text: Consider the modified Korteweg-de Vries equation (mKdV): \[ z_t-6z^2z_x+z_{xxx}=0. \] By a straightforward calculation, we find the Bäcklund transformation for the mKdV equation in the form \[ (z+y)_x=(z-y)\sqrt{\mu^2+(z+y)^2}. \] It is easily verified that this transformation can be reduced to the classical one.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35A30 Geometric theory, characteristics, transformations in context of PDEs
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[1] S. I. Svinolupov and V. V. Sokolov, ”Vector-matrix generalizations of classical integrable equations,” Teoret. Mat. Fiz. 100(2), 214–218 (1994) [Theoret. and Math. Phys. 100 (2), 959–962 (1994)]. · Zbl 0875.35121
[2] M. Yu. Balakhnev ”On a class of integrable evolution vector equations” Teoret. Mat. Fiz. 142(1), 13–20 (2005) [Theoret. and Math. Phys. 142 (1), 8–14 (2005)]. · Zbl 1178.37068
[3] L. Bianchi, Lezioni di Geometria Differenziale (Spoerri, Pisa, 1902), Vol II.
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