# zbMATH — the first resource for mathematics

Construction of higher-dimensional nonlinear integrable systems and of their solutions. (English. Russian original) Zbl 0597.35115
Funct. Anal. Appl. 19, 89-101 (1985); translation from Funkts. Anal. Prilozh. 19, No. 2, 11-15 (1985).
On the basis of the nonlocal Riemann problem and nonlocal $${\bar \partial}$$-problem a systematic exposition is given of the procedure of constructing higher dimensional integrable nonlinear equations. The nonlocal Riemann problem is to find an $$N\times N$$ matrix function $$\chi$$ ($$\lambda)$$ analytic everywhere off the contour $$\Gamma$$, whose boundary values $$\chi_ 1$$ and $$\chi_ 2$$ on $$\Gamma$$ are connected by the integral relation $\chi_ 2(\lambda)=\chi_ 1(\lambda)+\int_{\Gamma}\chi_ 1(\lambda ')T(\lambda ',\lambda)d\lambda '.$ It is supposed that T($$\lambda$$,$$\lambda$$ ’) depends on $$n\geq 2$$ supplementary variables $$x_ i$$ in such a way that $\partial T(\lambda ',\lambda)/\partial x_ i=I_ i(\lambda ')T(\lambda ',\lambda)-T(\lambda ',\lambda)I_ i(\lambda),$ where $$I_ i(\lambda)$$ are pairwise-commuting matrix-valued rational functions of the parameter $$\lambda$$. It is shown that in this case the solution $$\chi$$ of the Riemann problem satisfies the system of linear differential equations generating the corresponding nonlinear equations. The suggested procedure of constructing nonlinear equations and their solutions remains the same if one uses the more generalized nonlocal $${\bar \partial}$$- problem $2i \partial \chi (\lambda,{\bar \lambda})/\partial {\bar \lambda}=\int \chi (\lambda ',{\bar \lambda}') R(\lambda ',{\bar \lambda}',\lambda,\quad {\bar \lambda}) d\lambda ' d{\bar \lambda}'$ where $$\partial R/\partial x_ i=I_ i(\lambda ') R(\lambda ',{\bar \lambda}',\lambda,{\bar \lambda})-R(\lambda ',{\bar \lambda}',\lambda,\lambda \bar {\;}) I_ i(\lambda).$$
The effectiveness of the developed methods are illustrated on the Kadomtsev-Petviashvili equation and the Korteweg-de Vries equation.
Reviewer: A.B.Borisov

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 35N15 $$\overline\partial$$-Neumann problems and formal complexes in context of PDEs
Full Text:
##### References:
 [1] V. E. Zakharov and A. B. Shabat, ”A scheme for the integration of nonlinear evolution equations of mathematical physics by the inverse scattering method. I,” Funkts. Anal. Prilozhen.,6, No. 3, 43-53 (1974). · Zbl 0303.35024 [2] V. E. Zahkarov and A. B. Shabat, ”Integration of nonlinear equations of mathematical physics by the inverse scattering method. II,” Funkts. Anal. Prilozhen.,13, No. 3, 13-22 (1970). [3] S. V. Manakov, ”The inverse scattering transform for time-dependent Schrödinger equation and Kadomtsev?Petviashvili equation,” Physica 3D,3, Nos. 1-2, 420-427 (1981). · Zbl 1194.35507 [4] S. V. Manakov, P. Santini, and L. A. Takhtajan, ”Long-time behavior of the solutions of the Kadomtsev?Petviashvili equation,” Phys. Lett.,74A, 451-454 (1980). [5] D. J. Kaup, ”The inverse scattering solution for the full three-dimensional three-wave resonant interaction,” Studies Appl. Math.,62, 75-83 (1980). · Zbl 0451.35099 [6] M. J. Ablowitz, D. Bar Yaacov, and A. S. Fokas, ”On the inverse scattering transform for the Kadomtsev?Petviashvili equation,” Studies Appl. Math.,69, 135-143 (1983). · Zbl 0527.35080 [7] M. J. Ablowitz and A. S. Fokas, ”On the inverse scattering for the time-dependent Schrödinger equation and the associated Kadomtsev?Petviashvili (I) equation,” Studies Appl. Math.,69, 211-228 (1983). · Zbl 0528.35079 [8] V. E. Zakharov, ”The inverse scattering method,” in: Solitons, R. K. Bullough and P. J. Caudrey (eds.), Springer-Verlag, Berlin (1980), pp. 243-286. [9] V. E. Zakharov, ”Integrable systems in multidimensional space,” in: Math. Problems in Theor. Physics, Lecture Notes in Phys., Vol. 153, Springer-Verlag, Berlin (1982), pp. 190-216. [10] V. E. Zakharov, ”Multidimensional integrable systems,” Report to the International Congress of Math., Warsaw (1983). · Zbl 0525.76090 [11] S. V. Manakov, ”Nonlocal Riemann problem and solvable multidimensional nonlinear equations,” talk delivered at the NORDITA-Landau Institute Workshop, Copenhagen, September (1982). [12] V. E. Zakharov and S. V. Manankov, ”Multidimensional nonlinear integrable systems and methods for constructing their solutions,” Zap. Nauchn. Sem. LOMI,133, 77-91 (1984). · Zbl 0553.35078 [13] V. E. Zakharov and A. V. Mikhailov, ”On the integrability of classical spinor models in two-dimensional space?time,” Commun. Math. Phys.,74, 4-40 (1980). · doi:10.1007/BF01197576 [14] V. E. Zakharov and A. V. Mikhailov, ”Variational principle for the equations integrable by the method of the inverse problem,” Funkts. Anal. Prilozhen.,14, No. 1, 55-56 (1980). [15] S. V. Manakov, ”The inverse scattering method and two-dimensional evolution equations,” Usp. Mat. Nauk,31, No. 5, 245 (1976). [16] S. K. Zhdanov and B. L. Trubnikov, ”Soliton chains in a plasma with magnetic viscosity,” Pis’ma Zh. Eksp. Teor. Fiz.,39, No. 3, 110-113 (1983). [17] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevski, Theory of Solitons [in Russian], Nauka, Moscow (1980).
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.