Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 006, 13 pages      arXiv:1206.3751      https://doi.org/10.3842/SIGMA.2013.006

On the N-Solitons Solutions in the Novikov-Veselov Equation

Jen-Hsu Chang
Department of Computer Science and Information Engineering, National Defense University, Tauyuan, Taiwan

Received October 01, 2012, in final form January 12, 2013; Published online January 20, 2013

Abstract
We construct the $N$-solitons solution in the Novikov-Veselov equation from the extended Moutard transformation and the Pfaffian structure. Also, the corresponding wave functions are obtained explicitly. As a result, the property characterizing the $N$-solitons wave function is proved using the Pfaffian expansion. This property corresponding to the discrete scattering data for $N$-solitons solution is obtained in [arXiv:0912.2155] from the $\overline\partial$-dressing method.

Key words: Novikov-Veselov equation; $N$-solitons solutions; Pfaffian expansion; wave functions.

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References

  1. Athorne C., Nimmo J.J.C., On the Moutard transformation for integrable partial differential equations, Inverse Problems 7 (1991), 809-826.
  2. Basalaev M.Yu., Dubrovsky V.G., Topovsky A.V., New exact solutions with constant asymptotic values at infinity of the NVN integrable nonlinear evolution equation via $\overline\partial$-dressing method, arXiv:0912.2155.
  3. Bogdanov L.V., Veselov-Novikov equation as a natural two-dimensional generalization of the Korteweg-de Vries equation, Theoret. Math. Phys. 70 (1987), 219-223.
  4. Chang J.H., The Gould-Hopper polynomials in the Novikov-Veselov equation, J. Math. Phys. 52 (2011), 092703, 15 pages, arXiv:1011.1614.
  5. Dubrovin B.A., Krichever I.M., Novikov S.P., The Schrödinger equation in a periodic field and Riemann surfaces, Sov. Math. Dokl. 17 (1976), 947-952.
  6. Dubrovsky V.G., Formusatik I.B., New lumps of Veselov-Novikov integrable nonlinear equation and new exact rational potentials of two-dimensional stationary Schrödinger equation via $\overline\partial$-dressing method, Phys. Lett. A 313 (2003), 68-76.
  7. Dubrovsky V.G., Formusatik I.B., The construction of exact rational solutions with constant asymptotic values at infinity of two-dimensional NVN integrable nonlinear evolution equations via the $\overline\partial$-dressing method, J. Phys. A: Math.Gen. 34 (2001), 1837-1851.
  8. Grinevich P.G., Rational solitons of the Veselov-Novikov equations are reflectionless two-dimensional potentials at fixed energy, Theoret. Math. Phys. 69 (1986), 1170-1172.
  9. Grinevich P.G., Manakov S.V., Inverse scattering problem for the two-dimensional Schrödinger operator, the $\overline\partial$-method and nonlinear equations, Funct. Anal. Appl. 20 (1986), 94-103.
  10. Grinevich P.G., Mironov A.E., Novikov S.P., New reductions and nonlinear systems for 2D Schrödinger operators, arXiv:1001.4300.
  11. Hirota R., The direct method in soliton theory, Cambridge Tracts in Mathematics, Vol. 155, Cambridge University Press, Cambridge, 2004.
  12. Hu H.C., Lou S.Y., Construction of the Darboux transformaiton and solutions to the modified Nizhnik-Novikov-Veselov equation, Chinese Phys. Lett. 21 (2004), 2073-2076.
  13. Hu H.C., Lou S.Y., Liu Q.P., Darboux transformation and variable separation approach: the Nizhnik-Novikov-Veselov equation, Chinese Phys. Lett. 20 (2003), 1413-1415, nlin.SI/0210012.
  14. Ishikawa M., Wakayama M., Applications of minor-summation formula. II. Pfaffians and Schur polynomials, J. Combin. Theory Ser. A 88 (1999), 136-157.
  15. Kaptsov O.V., Shan'ko Yu.V., Trilinear representation and the Moutard transformation for the Tzitzéica equation, solv-int/9704014.
  16. Kodama Y., KP solitons in shallow water, J. Phys. A: Math. Gen. 43 (2010), 434004, 54 pages, arXiv:1004.4607.
  17. Kodama Y., Maruno K., $N$-soliton solutions to the DKP equation and Weyl group actions, J. Phys. A: Math. Gen. 39 (2006), 4063-4086, nlin.SI/0602031.
  18. Kodama Y., Williams L.K., KP solitons and total positivity for the Grassmannian, arXiv:1106.0023.
  19. Kodama Y., Williams L.K., KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. USA 108 (2011), 8984-8989, arXiv:1105.4170.
  20. Konopelchenko B.G., Introduction to multidimensional integrable equations. The inverse spectral transform in 2+1 dimensions, Plenum Press, New York, 1992.
  21. Konopelchenko B.G., Landolfi G., Induced surfaces and their integrable dynamics. II. Generalized Weierstrass representations in 4D spaces and deformations via DS hierarchy, Stud. Appl. Math. 104 (2000), 129-169.
  22. Krichever I.M., A characterization of Prym varieties, Int. Math. Res. Not. 2006 (2006), Art. ID 81476, 36 pages, math.AG/0506238.
  23. Liu S.Q., Wu C.Z., Zhang Y., On the Drinfeld-Sokolov hierarchies of $D$ type, Int. Math. Res. Not. 2011 (2011), 1952-1996, arXiv:0912.5273.
  24. Manakov S.V., The method of the inverse scattering problem, and two-dimensional evolution equations, Russian Math. Surveys 31 (1976), no. 5, 245-246.
  25. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
  26. Mironov A.E., A relationship between symmetries of the Tzitzéica equation and the Veselov-Novikov hierarchy, Math. Notes 82 (2007), 569-572.
  27. Mironov A.E., Finite-gap minimal Lagrangian surfaces in ${\mathbb C}{\rm P}^2$, in Riemann Surfaces, Harmonic Maps and Visualization, OCAMI Stud., Vol. 3, Osaka Munic. Univ. Press, Osaka, 2010, 185-196, arXiv:1005.3402.
  28. Mironov A.E., The Veselov-Novikov hierarchy of equations, and integrable deformations of minimal Lagrangian tori in ${\mathbb C}{\rm P}^2$, Sib. Electron. Math. Rep. 1 (2004), 38-46, math.DG/0607700.
  29. Moutard M., Note sur les équations différentielles linéaires du second ordre, C.R. Acad. Sci. Paris 80 (1875), 729-733.
  30. Moutard M., Sur la construction des équations de la forme $\frac{1}{z} \frac{\partial^2z}{\partial x\partial y} =\lambda(xy)$, qui admettent une intégrale générale explicite, J. de. l'Éc. Polyt. 28 (1878), 1-12.
  31. Nimmo J.J.C., Darboux transformations in (2+1)-dimensions, in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (Exeter, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 413, Kluwer Acad. Publ., Dordrecht, 1993, 183-192.
  32. Nimmo J.J.C., Hall-Littlewood symmetric functions and the BKP equation, J. Phys. A: Math. Gen. 23 (1990), 751-760.
  33. Novikov S.P., Two-dimensional Schrödinger operators in periodic fields, J. Sov. Math. 28 (1985), 1-20.
  34. Novikov S.P., Veselov A.P., Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations, Phys. D 18 (1986), 267-273.
  35. Ohta Y., Pfaffian solutions for the Veselov-Novikov equation, J. Phys. Soc. Japan 61 (1992), 3928-3933.
  36. Orlov A.Yu., Shiota T., Takasaki K., Pfaffian structures and certain solutions to BKP hierarchies. I. Sums over partitions, arXiv:1201.4518.
  37. Shiota T., Prym varieties and soliton equations, in Infinite-Dimensional Lie Algebras and Groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 407-448.
  38. Stembridge J.R., Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83 (1990), 96-131.
  39. Taimanov I.A., Tsarev S.P., Two-dimensional rational solitons and their blowup via the Moutard transformation, Theoret. Math. Phys. 157 (2008), 1525-1541, arXiv:0801.3225.
  40. Takasaki K., Dispersionless Hirota equations of two-component BKP hierarchy, SIGMA 2 (2006), 057, 22 pages, nlin.SI/0604003.
  41. Veselov A.P., Novikov S.P., Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations, Sov. Math. Dokl. 30 (1984), 588-591.

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