## $$N$$-soliton strings in four-dimensional space-time.(English. Russian original)Zbl 1170.83469

Theor. Math. Phys. 152, No. 3, 1234-1242 (2007); translation from Teor. Mat. Fiz. 152, No. 3, 430-439 (2007).
Summary: We investigate infinite relativistic strings in the Minkowski space $$E_{1,3}$$ set theoretically. We show that the set of such strings is uniquely parameterized by elements of the Poincaré group $$\mathcal{P}$$, of the group $$\mathcal{D}$$ of scaling transformations of Minkowski space, and of a certain subgroup $$\mathcal{W}_0$$ of the group of Weyl transformations of the two-metric and also by elements of the set of scattering data for a pair of first-order spectral problems that are characteristic of the theory of the nonlinear Schrödinger equation. The coefficients of the spectral problems are related to the second quadratic forms of the worldsheet. In this context, we define $$N$$-soliton strings. We discuss a hierarchy of surfaces that occurs in this analysis and corresponds to the known hierarchy associated with the nonlinear Schrödinger equation.

### MSC:

 83E30 String and superstring theories in gravitational theory 55Q51 $$v_n$$-periodicity 83A05 Special relativity
Full Text:

### References:

 [1] M. R. Anderson, The Mathematical Theory of Cosmic Strings: Cosmic Strings in the Wire Approximation, IOP Publ., Bristol (2003). · Zbl 1010.83005 [2] V. A. Rubakov, Phys. Usp., 44, 871 (2001). [3] S. V. Talalov, Theor. Math. Phys., 123, 446 (2000). · Zbl 1031.81629 [4] S. V. Talalov, J. Phys. A, 22, 2275 (1989). · Zbl 0701.58060 [5] B. M. Barbashov and V. V. Nesterenko, Model of Relativistic Strings in the Physics of Hadrons [in Russian], Enrgoatomizdat, Moscow (1987); English transl.: Introduction to Relativistic String Theory, World Scientific, Singapore (1990). [6] S. P. Novikov, Soviet Math. Dokl., 24, 222–226 (1981). [7] E. Witten, Comm. Math. Phys., 92, 455 (1984). · Zbl 0536.58012 [8] A. K. Pogrebkov and S. V. Talalov, Theor. Math. Phys., 70, 241 (1987). [9] A. K. Pogrebkov, Theor. Math. Phys., 12, 765 (1972). [10] G. P. Pron’ko, A. V. Razumov, and L. D. Solov’ev, Sov. J. Part. Nucl., 14, No. 3, 229–237 (1983). [11] L. A. Takhtadzhan and L. D. Faddeev, Hamiltonian Approach in the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl.: L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Berlin, Springer (1987). · Zbl 0618.35100 [12] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevsky, Theory of Solitons: Method of the Inverse Problem [in Russian], Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Plenum, New York (1984). · Zbl 0598.35003 [13] S. V. Talalov, Theor. Math. Phys., 71, 588 (1987). · Zbl 0649.35078 [14] S. V. Klimenko and I. N. Nikitin, Theor. Math. Phys., 114, 299 (1998). · Zbl 0956.53044 [15] P. P. Kulish and A. G. Rejman, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 77, 134 (1978).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.