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Applicable Analysis
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Volume 95, 2016 - Issue 6
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Articles

The interactions of solitons in the Novikov–Veselov equation

Pages 1370-1388 | Received 25 Mar 2015, Accepted 16 Jun 2015, Published online: 13 Jul 2015

References

  • Bogdanov LV. The Veselov-Novikov equation as a natural two-dimensional generalization of the KdV equation. Ther. Math. Fiz. 1987;70:309–314.
  • Grinevich PG, Manakov SV. Inverse scattering problem for the two-dimensional Schrödinger operator, the ∂̄-method and nonlinear equations. Funct. Anal. Appl. 1986;20:94–103.
  • Novikov SP, Veselov AP. Two-dimensional Schrödinger operator: Inverse scattering transform and evolutional equations. Physica D. 1986;18:267–273.
  • Veselov AP, Novikov SP. Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations. Sov. Math. Dokl. 1984;30:588–591.
  • Manakov SV. The method of the inverse scattering problem, and two-dimensional evolution equations. Uspekhi Mat. Nauk. 1976;31:245–246.
  • Dubrovin BA, Krichever IM, Novikov SP. The Schrödinger equation in a periodic field and Riemann surfaces. Soviet Math. Dokl. 1976;17:947–952.
  • Grinevich P, Mironov A, Novikov S. New reductions and nonlinear systems for 2D Schrödinger operators. 2010, arXiv:1001.4300.
  • Krichever I. A characterization of Prym varieties. Int. Math. Res. Notices. 2006;2006: Art. ID 81476, 36 p, doi:10.1155/IMRN/2006/81476
  • Novikov SP. Two-dimensional Schrödinger operators in periodic fields. J. Sov. Math. 1985;28:1–20.
  • Shiota T. Prym varieties and soliton equations. In: Victor G. Kac, editor. Infinite-dimensional Lie algebras and groups, (Luminy-Marseille, 1988). Vol. 7, Advance series in mathematical physics. Teaneck (NJ): World Scientific Publications; 1989. p. 407–448.
  • Liu S-Q, Wu C-Z, Zhang Y. On the Drinfeld-Sokolov hierarchies of D type, Intern. Math. Res. Notices. 2011;8:1952–1996.
  • Takasaki K. Dispersionless Hirota equations of two-component BKP hierarchy. SIGMA. 2006;2. Paper 057. 22 p. arXiv:nlin.SI/0604003.
  • Konopelchenko BG. Introduction to multidimensional integrable equations: the inverse spectral transform in 2+1-dimensions. New York (NY): Plenum Press; 1992.
  • Konopelchenko BG, Landolfi G. Induced surfaces and their integrable dynamics II: generalized Weierstrass representation in 4-d spaces and deformations via DS hierarchy. Stud. Appl. Math. 2000;104:129–169.
  • Athorne C, Nimmo JJC. On the Moutard transformation for integrable partial differential equations. Inverse Probl. 1991;7:809–826.
  • Moutard Th--F. Note sur les équations différentielles linéaires au second ordre [Note on the linear second order differential equation], C.R. Acad Sci (Paris), 1875;8080:729–733.
  • Nimmo JJC. Darboux transformations in (2 + 1) dimensions. In: Clarkson P, editors. Proc. NATO ARW, Applications of analytic and geometric methods to nonlinear differential equations, workshop held at the University of Exeter, U.K, 1992, Vol. 192, NATO ASI Series. Kluwer; 1993. p. 183–192.
  • Hu H-C, Lou S-Y, Liu Q-P. Darboux transformation and variable separation approach: the Nizhnik--Novikov--Veselov equation. Chin. Phys. Lett. 2003;20:1413–1415.
  • Matveev VB, Salle MA. Darboux transformations and solitons. Springer series in nonlinear dynamics. Berlin: Springer; 1991.
  • Hu H-C, Lou S-Y. Construction of the Darboux transformation and solutions to the modified Nizhnik--Novikov--Veselov equation. Chin. Phys. Lett. 2004;21:2073–2076.
  • Ohta Y. Pfaffian solution for the Veselov-Novikov equation. J. Phys. Soc. Jpn. 1992;61:3928–3933.
  • Ablowitz Mark J, Baldwin Douglas E. Nonlinear shallow ocean wave soliton interactions on flat beaches. Phys. Rev. E. 2012;86:036305.
  • Biondini G, Chakravarty S. Soliton solutions of the Kadomtsev--Petviashvili II equation. J. Math. Phys. 2006;47:033514, arXiv:nlin/0511068.
  • Chakravarty S, Lewkow T, Maruno K-I. On the construction of the KP line-solitons and their interactions. Appl. Anal. 2010;89:529–545, arXiv: 0911.2290.
  • Kodama Y, Williams L. KP solitons and total positivity for the Grassmannian, Invent. Math. 2014;198:637–699, arXiv:1106.0023.
  • Kodama Y, Williams L. KP solitons and total positivity for the Grassmannian, Invent. Math. 2014;198:637–699, arXiv:1106.0023.
  • Kodama Y, Williams LK. The Deodhar decomposition of the Grassmannian and the regularity of KP solitons. Adv. Math. 2013;244:979–1032, arXiv:1204.6446.
  • Dubrovsky1 VG, Topovsky AV. About linear superpositions of special exact solutions of Nizhnik--Veselov--Novikov equation. J. Phys.: Conf. Ser. 2014;482:012011; J. Phys.: Conf. Ser. 2013. PMNP2013.
  • Ishikawa M, Wakayama M. Applications of minor summation formulas II, Pfaffians of Schur polynomials. J. Combin. Theory Ser. A. 1999;88:136–157.
  • Kodama Y, Maruno K. N-soliton solitons to the DKP equation and Weyl group actions. J. Phys. A: Math. Gen. 2006;39:4063–4086, arXiv:nlin/0602031.
  • Kodama Y, Williams L. KP solitons, total positivity, and cluster algebras, Proceedings of the National Academy of Sciences of the United States of America (PNAS); 2011;108:8984–8989, arXiv:1105.4170.
  • Basalaev MY, Dubrovsky VG, Topovsky AV. New exact multi line soliton and periodic solutions with constant asymptotic values at infinity of the NVN integrable nonlinear evolution equation via ∂̄-dressing method. 2009; 43, arXiv: 0912.2155. (preprint).
  • Dubrovsky V, Topovsky AV, Basalaev MY. New exact solutions of two-dimensional integrable equations using the ∂̄-dressing method. Theor. Math. Phys. 2011;167:725–739.
  • Chang J-H. N-solitons solutions in the Novikov–Veselov equation. SIGMA. 2013;9:006. 13 p. arXiv:1206.3751.
  • Grinevich PG. Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity. Russ. Math. Surv. 2000;55:1015–1083.
  • Nimmo JJC. Hall-Littlewood symmetric functions and the BKP equation. J. Phys. A : Math. Gen. 1990;23:751–760.
  • Orlov AY, Shiota T, Takasaki K. Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions, arXiv:1201.4518. (preprint).
  • Miles JW. Resonantly interacting solitary waves. J. Fluid Mech. 1977;79:171–179.
  • Chakravarty S, Kodama Y. Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Math. 2009;123:83–51, arXiv: 0902.4423.

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