# The Direct Method In Soliton Theory

This summer I'm going to learn a mini-course about soliton theory ("Soliton equations and symmetric functions" in LHSM (Russian summer school in mathematics).The web-page of this course is (warning: it's in Russian, so here's my translation of its program:

## The Direct Method in Soliton Theory

This program has intrigued me (despite my main interest is representation theory and related topics) and now I'm looking for a book in which the soliton theory will be outlined according to this program (i.e. using Hirota derivatives from very beginning), maybe even without any mentions of symmetric functions..

This paper presents the q-analogue of Toda lattice system of difference equations by discussing the q-discretization in three aspects: differential-q-difference, q-difference-q-difference and q-differential-q-difference Toda equation. The paper develops three-q-soliton solutions, which are expressed in the form of a polynomial in power functions, for the differential-q-difference and q-difference-q-difference Toda equations by Hirota direct method. Furthermore, it introduces q-Hirota D-operator and presents the q-differential-q-difference version of Toda equation. Finally, the paper presents its solitary wave like a solution in terms of q-exponential function and explains the nonexistence of further solutions in terms of q-exponentials by the virtue of Hirota perturbation.

Inspired by this fact, the purpose of this paper is to present the q-analogue of Toda lattice system of difference equations and discuss the applicability of Hirota direct method for constructing multi-soliton solutions. There are several ways to q-discretize a given continuous equation. By q-discretization we mean q-difference equations determined by q-difference operator and additionally q-differential equations constructed by q-derivative operator. Therefore, we present q-discretization in three aspects: differential-q-difference, q-difference-q-difference and q-differential-q-difference Toda equation. We show that Hirota direct method allows to produce three-soliton solutions for the differential-q-difference and q-difference-q-difference Toda equations. We emphasize that the solutions not only possess soliton behaviors but have additional power counterparts for q-discrete variables. We call such soliton solutions as q-soliton solutions. Therefore, unlike the Toda equation [14] or discrete-time Toda equation [15], the differential-q-difference and q-difference-q-difference Toda equations have soliton solutions in the form of a polynomial in power functions. On the other hand, we conclude that Hirota direct method fails to derive multi-soliton solutions of the q-differential-q difference Toda equation. Furthermore, it is not possible to obtain multi-soliton solutions for any q-differential-q-difference or q-differential-difference type of equations by means of Hirota perturbation.

This form is called bilinear form of F[u]. We should remark that some integrable equations can only be transformed to a single bilinear form while some of them can be written as a combination of bilinear forms. On the other hand, for some equations it is not possible to find a proper transformation. The next step towards Hirota direct method, is introducing the so-called Hirota D-operator which is a binary differential operator exhibiting a new calculus.

To be more precise, although the equation (82) can be put into Hirota bilinear form (83), in both cases Hirota perturbation fails to produce further solutions. Moreover, it is straightforward to conclude that for any q-differential-q-difference or q-differential-difference type of equation even if the equation has a Hirota bilinear form, it is not possible to derive multisoliton solutions by the use of Hirota Direct method.

The structure of this paper is as follows. In the second section, some basic knowledge on time-space scale are introduced. The third section is important that new AKNS system is constructed, and specific parameters are selected to obtain the KdV equation on time-space scale, which can be simplified into classical and discrete KdV equation. In Section 4, the single-soliton solution of KdV equation under the time scale framework is constructed by using the idea of direct method, and the nonlinear dispersion relationship of the equation is obtained. In particular, solutions of KdV equation on two different time scales are obtained. The last part is our conclusion.

In this paper, a method of generating integrable system on time-space scale is introduced. Starting from the -dynamical system, the coupled KdV equation on time-space scale is derived from the Lax pair and zero curvature equation. When different time scales are considered, different soliton equations can be obtained. In addition, the variable transformation of the KdV equation on time-space scale is constructed to obtain its single-soliton solution.

Solitons are an important class of solutions to nonlinear differential equations which appear in different areas of physics and applied mathematics. In this study we provide a general overview of the Hirota method which is one of the most powerful tool in finding the multi-soliton solutions of nonlinear wave and evaluation equations. Bright and dark soliton solutions of nonlinear Schrödinger equation are discussed in detail

In Soliton theory, Hirota direct method is most efficient tool for seeking one soliton solutions or multi-soliton solutions of integrable nonlinear partial differential equations. The key step of the Hirota direct method is to transform the given equation into its Hirota bilinear form. Once the bilinear form of the given equation is found, we can construct the soliton and multi-soliton solutions of that model. Many interesting characteristics of Pfaffians were discovered through studies of soliton equations. In this thesis, a shallow water wave model and its bilinear equation are investigated. Using Hirota direct method, we obtain the multi-soliton solutions and Pfaffian solutions for a shallow water wave model.

AbstractWe construct the $N$-solitons solution in the Novikov-Veselovequation from the extended Moutard transformation and the Pfaffianstructure. Also, the corresponding wave functions are obtainedexplicitly. 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.

ReferencesAthorne C., Nimmo J.J.C., On the Moutard transformation for integrable partial differential equations, Inverse Problems 7 (1991), 809-826.

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.

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.

Chang J.H., The Gould-Hopper polynomials in the Novikov-Veselov equation, J. Math. Phys. 52 (2011), 092703, 15 pages, arXiv:1011.1614.

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.

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.

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.

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.

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.

Grinevich P.G., Mironov A.E., Novikov S.P., New reductions and nonlinear systems for 2D Schrödinger operators, arXiv:1001.4300.

Hirota R., The direct method in soliton theory, Cambridge Tracts in Mathematics, Vol. 155, Cambridge University Press, Cambridge, 2004.

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.

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.

Ishikawa M., Wakayama M., Applications of minor-summation formula. II. Pfaffians and Schur polynomials, J. Combin. Theory Ser. A 88 (1999), 136-157.

Kaptsov O.V., Shan'ko Yu.V., Trilinear representation and the Moutard transformation for the Tzitzéica equation, solv-int/9704014.

Kodama Y., KP solitons in shallow water, J. Phys. A: Math. Gen. 43 (2010), 434004, 54 pages, arXiv:1004.4607.

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.

Kodama Y., Williams L.K., KP solitons and total positivity for the Grassmannian, arXiv:1106.0023.

Kodama Y., Williams L.K., KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. USA 108 (2011), 8984-8989, arXiv:1105.4170.

Konopelchenko B.G., Introduction to multidimensional integrable equations. The inverse spectral transform in 2+1 dimensions, Plenum Press, New York, 1992.

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.

Krichever I.M., A characterization of Prym varieties, Int. Math. Res. Not. 2006 (2006), Art. ID 81476, 36 pages, math.AG/0506238.

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.

Manakov S.V., The method of the inverse scattering problem, and two-dimensional evolution equations, Russian Math. Surveys 31 (1976), no. 5, 245-246.

Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.

Mironov A.E., A relationship between symmetries of the Tzitzéica equation and the Veselov-Novikov hierarchy, Math. Notes 82 (2007), 569-572.

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.

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.

Moutard M., Note sur les équations différentielles linéaires du second ordre, C.R. Acad. Sci. Paris 80 (1875), 729-733.

Moutard M., Sur la construction des équations de la forme $\frac1z \frac\partial^2z\partial x\partial y =\lambda(xy)$, qui admettent une intégrale générale explicite, J. de. l'Éc. Polyt. 28 (1878), 1-12.

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.

Nimmo J.J.C., Hall-Littlewood symmetric functions and the BKP equation, J. Phys. A: Math. Gen. 23 (1990), 751-760.

Novikov S.P., Two-dimensional Schrödinger operators in periodic fields, J. Sov. Math. 28 (1985), 1-20.

Novikov S.P., Veselov A.P., Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations, Phys. D 18 (1986), 267-273.

Ohta Y., Pfaffian solutions for the Veselov-Novikov equation, J. Phys. Soc. Japan 61 (1992), 3928-3933.

Orlov A.Yu., Shiota T., Takasaki K., Pfaffian structures and certain solutions to BKP hierarchies. I. Sums over partitions, arXiv:1201.4518.

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.

Stembridge J.R., Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83 (1990), 96-131.

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.

Takasaki K., Dispersionless Hirota equations of two-component BKP hierarchy, SIGMA 2 (2006), 057, 22 pages, nlin.SI/0604003.

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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