A class of solutions of the two-dimensional Toda lattice equation
aa r X i v : . [ n li n . S I] S e p A class of solutions of the two-dimensional Toda lattice equation
V. N. Duarte ∗ Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ, 08543, USA (Dated: September 9, 2020)A method is proposed to systematically generate solutions of the two-dimenional Toda latticeequation in terms of previously known solutions φ ( x, y ) of the two-dimensional Laplace’s equation.The two-dimensional solution of Nakamura’s [J. Phys. Soc. Jpn. , 380 (1983)] is shown tocorrespond to one particular choice of φ ( x, y ). The numerical discovery of solitons in collisionless plas-mas [1], accompanied by an explanation for the recur-rence of states in nonlinear prototype string systems [2],and the subsequent development of the inverse scatter-ing tranform to integrate the Korteweg-de Vries equa-tion [3, 4] led to an intense search for exactly solv-able, completely integrable systems described by disper-sive nonlinear partial differential equations. More mod-ern applications of such systems include topological soli-tons in the context of magnetic skyrmions [5] and theWess–Zumino–Witten model [6, 7].Among the mentioned class of exactly solvable sys-tems is Toda’s devise of a potential form that couplesnearest neighbors in a lattice that allows for completeintegration of the equation that governs the vibrationsof the lattice structure [8]. In the continuum limit, theone-dimensional Toda equation recovers the Korteweg-deVries equation [9] while the two-dimensional Toda equa-tion recovers [10] the Kadomtsev-Petviashvili equation,originally derived to model the effect of long transverseperturbations on the dynamics of plasma ion-acousticmodes of long wavelength and small amplitude [11].For many nonlinear problems, it is useful to have theirsolutions expressed in terms of solutions to linear prob-lems, as is the case of approaches that express the so-lutions of Liouville’s equation in terms of solutions ofLaplace’s equation [12, 13] and the case of the Cole-Hopftransform [14, 15] that maps solutions of the Burgers’equation onto solutions of a linear diffusion equation. In-spired by the former, this note develops a technique toexpress solutions, both solitonic and non-solitonic, of thetwo-dimensional Toda lattice equation (with two contin-uous variables
X, Y and one discrete variable Z ≡ n )in terms of solutions of Laplace’s equation, for whichexistence and uniqueness of solutions are guaranteedfor given boundary conditions. The two-dimensionalToda lattice equation with its corresponding exponentialrestoring force reads [9, 16] αu XX ( X, Y, n ) + βu Y Y ( X, Y, n ) == e u ( X,Y,n − − u ( X,Y,n ) − e u ( X,Y,n ) − u ( X,Y,n +1) . (1)To exploit the properties of Laplace’s equation, the con- tinuous variables will be rescaled as x ≡ X/ √ α, y ≡ Y / p β, (2)hence u xx ( x, y, n ) + u yy ( x, y, n ) == e u ( x,y,n − − u ( x,y,n ) − e u ( x,y,n ) − u ( x,y,n +1) . (3)Solutions to Eq. (3) will be sought using the ansatz u ( x, y, n ) = F ( φ ( x, y ) , n ) + ln h |∇ φ ( x, y ) | − n i , (4)where φ ( x, y ) is a solution of the two-dimensionalLaplace’s equation φ xx + φ yy = 0 (5)and F ( φ, n ) is a function to be determined. The presentderivation takes advantage of the property noted byClemente [13] that, if (5) holds, then also[ln ( |∇ φ | γ )] xx + [ln ( |∇ φ | γ )] yy = 0 (6)is true ∀ γ ∈ C . In addition, the calculation will usethe fact that, for any function ψ ≡ ψ ( φ ( x, y ) , n ), therelation [17] ∇ ψ = ∂ψ∂φ ∇ φ + ∂ ψ∂φ |∇ φ | (7)holds as long as the Laplacian and the gradient are two-demensional in x, y and therefore independent of the vari-able n . Subtituting the ansatz (4) into Eq. (3) and sim-plifying the resulting expression using Eqs. (5), (6) and(7), the dimensionality of the problem is reduced from( x, y, n ) to ( φ, n ) and Eq. (3) becomes ∂ ∂φ F ( φ ( x, y ) , n ) = e F ( φ ( x,y ) ,n − − F ( φ ( x,y ) ,n ) −− e F ( φ ( x,y ) ,n ) − F ( φ ( x,y ) ,n +1) , (8)Note that the ansatz (4) was chosen in such a way asto exactly cancel out the dependencies on |∇ φ | that oth-erwise would be present in Eq. (8). The fact that for-mally Eq. (8) only depends on two independent variables,rather than three as is the case of Eqs. (1) and (3), allowsfor a simplified construction of conservation laws relativeto Ref. [18]. Eq. (8) can be expressed in terms of thefunction r ( φ, n ) ≡ F ( φ, n ) − F ( φ, n −
1) as ∂ r ( φ, n ) ∂φ = 2 e r ( φ,n ) − e r ( φ,n − − e r ( φ,n +1) (9)Defining f ( φ, n ) via e − F ( φ,n )+ F ( φ,n − = e − r ( φ,n ) = c + ∂ ∂φ ln f ( φ, n ) , (10)with c being a constant, allows Eq. (9) to be cast inHirota’s bilinear form [19], following a similar procedureas done in the one-dimensional case [9]. The substitutionof Eq. (10) into Eq. (9) leads to e − F ( φ,n )+ F ( φ,n − = f ( φ, n + 1) f ( φ, n − f ( φ, n ) e c φ + c (11)where c and c are constants of integration. A solutioncan then be obtained through the ansatz f ( φ, n ) = 1 + e pφ +¯ ωn , (12)which yields e − F ( φ,n )+ F ( φ,n − = (cid:20) p sech (cid:18) pφ + ¯ ωn (cid:19)(cid:21) e c φ + c , (13)where sinh (¯ ω/
2) = ± p/
2. Or, in terms of the originalfunction as defined by (4), e − u ( x,y,n )+ u ( x,y,n − == (cid:20) p sech (cid:18) pφ + ¯ ωn (cid:19)(cid:21) e c φ + c |∇ φ | . (14)The one-soliton solution of Ref. [16] is exactly re-covered with the particular choice of a traveling waveform for the Laplace’s equation solution φ , linear inboth x and y , specifically φ ( x, y ) = (cid:0) √ αkx + √ βly (cid:1) /p with p = ± p αk + βl , c = c = 0. In that case |∇ φ ( x, y ) | = 1. Using these choices and returning tothe orginal variables using Eq. (2), Eq. (14) becomes e − u ( X,Y,n )+ u ( X,Y,n − − αk + βl sech [( kX + lY + ¯ ωn ) / . (15)Eq. (15) is the same as Eq. 3.8 of Nakamura’s paper [16].It can be appreciated that the most stringent assumptionof the present derivation was to assume a form for f (Eq.(12)). It should be noted, however, that other solutions,more convenient for a given physical problem at hand, can be constructed for different forms of f but still usingthe same methodology presented in this work, i.e., differ-ent f will lead to distinct F ( φ, n ) while still preservingthe same solution ansatz (Eq. (4)). Because of its e c φ ( c ∈ C ) dependence, Eq. (14) provides a direct meansfor constructing breather-like solutions [20], i.e., solutionswith localization due to amplitude decay in the contin-uous variables and oscillation in the discrete variable orvice-versa, depending on the choice of the constants c , p and ¯ ω .In summary, using the serendipitous identity (6) andan ansatz (4) that takes advantage of the structure ofthe underlying equation (3), new solutions of the two-dimensional (nonlinear) Toda lattice equation were con-structed in terms of solutions φ of the (linear) Laplace’sequation. A particular choice of φ was shown to repli-cate a previously found solution for the lattice equation[16]. Other solutions can then be constructed from (14)to the limits of the imposed boundary conditions of aparticular problem. For example, expression (14) pro-vides a rapid means for construction of solutions withazimuthal symmetry. In that case, (1 /r ) ( rφ r ) r = 0,where r = p X /α + Y /β and the sub-index denotesdifferentiation. Therefore any φ ∝ ln r in Eq. (14) willlead to an azimuthally invariant solution.Discussions with V. L. Quito are appreciated. Thiswork was supported by the US Department of Energyunder contract DE-AC02-09CH11466. ∗ [email protected][1] N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. ,240 (1965).[2] E. Fermi, P. Pasta, S. Ulam, and M. Tsingou, Studiesof nonlinear problems , Tech. Rep. (Los Alamos ScientificLab., NM, 1955).[3] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M.Miura, Phys. Rev. Lett. , 1095 (1967).[4] P. D. Lax, Comm. Pure Appl. Math. , 467 (1968).[5] A. N. Bogdanov and C. Panagopoulos, Nature Rev. Phys., 1 (2020).[6] J. Wess and B. Zumino, Phys. Lett. B , 95 (1971).[7] E. Witten, Nucl. Phys. B , 422 (1983).[8] M. Toda, J. Phys. Soc. Jpn. , 431 (1967).[9] M. Toda, Theory of Nonlinear Lattices , Vol. 20 (SpringerScience & Business Media, 2012).[10] R. Hirota, J. Phys. Soc. Jpn. , 3785 (1981).[11] B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. Dokl , , 539 (1970).[12] G. W. Walker, Proc. Royal Soc. London A , 410 (1915).[13] R. A. Clemente, Int. J. Math. Educ. Sci. Technol. , 620(1992).[14] J. D. Cole, Quart. Appl. Math. , 225 (1951).[15] E. Hopf, Comm. Pure Appl. Math. , 201 (1950).[16] A. Nakamura, J. Phys. Soc. Jpn. , 380 (1983).[17] This property has been extensively used in tokamakplasma modeling, to formally construct solutions for the poloidal flux stream function ψ of a pressure-anisotropicaxisymmetric equilibrium in terms of solutions of theisotropic Grad-Shafranov equation [21]. Such techniqueinvolves an integral transform and can also be employedin the presence of plasma rotation [22].[18] K. Kajiwara and J. Satsuma, J. Math. Phys. , 506 (1991).[19] R. Hirota, Progr. Theor. Phys. , 1498 (1974).[20] N. Flytzanis, S. Pnevmatikos, and M. Remoissenet, J.Phys. C , 4603 (1985).[21] R. A. Clemente, Nucl. Fusion , 963 (1993).[22] R. A. Clemente, Plasma Phys. Control. Fusion36