Discrete mKdV equation via Darboux transformation
DDISCRETE MKDV EQUATION VIA DARBOUXTRANSFORMATION
JOSEPH CHO, WAYNE ROSSMAN, AND TOMOYA SENO
Abstract.
We introduce an efficient route to obtaining the discrete potentialmKdV equation emerging from a particular discrete motion of discrete planarcurves. Introduction
In this work, we illuminate the relationship between discrete and semi-discretepotential modified Korteweg-de Vries (mKdV) equation on the one hand, and onthe other, the transformation theory of the smooth potential mKdV equation via theuse of infinitesimal Bianchi cubes of Darboux transformations keeping arc-lengthpolarization.Studies of the discrete and semi-discrete potential mKdV equation, starting witha discrete curve having an associated discrete frame, can be found in [4, 5, 6, 7].There discrete or continuous isoperimetric deformation was used to produce certainmotions of a curve, and then the discrete frames provided the means to produce thedesired equations. In fact, the discrete and semi-discrete potential mKdV equationappear in the context of the B¨acklund transformation and permutability of smoothpotential mKdV equations [8].To exploit this, we consider a 3-dimensional system with two discrete parametersand one smooth parameter, where the discrete parameters represent a Darbouxtransformation of a discrete curve, while the smooth parameter represents theinfinitesimal Darboux transformation [3]. With this combination of discrete andsmooth parameters, we can then take advantage of the result in [2] regardingthe permutability between Darboux and infinitesimal Darboux transformations(infinitesimal Bianchi cube). By restricting to the Darboux transformations thatpreserve arc-length polarization, we can reinterpret this system in terms of thetransformation and permutability of smooth potential mKdV equations, giving us acalculation-free approach to producing the fully discrete potential mKdV equationwithout any use of frames (see the primary result, Theorem 3.2). We note that suchpursuit is greatly aided by interpreting the potential functions appearing in boththe discrete and semi-discrete equations as a single geometric quantity encoded inthe system, as done in [3].
Mathematics Subject Classification.
Primary 53A70; Secondary 35Q53, 53A04. a r X i v : . [ n li n . S I] D ec JOSEPH CHO, WAYNE ROSSMAN, AND TOMOYA SENO Darboux and Infinitesimal Darboux transformations
Let Σ ⊂ Z be a discrete interval, and let µ be a strictly positive or negative functionon (unoriented) edges of Σ. Recall from [3], that a discrete polarized curve is adiscrete curve x : (Σ , µ ) → C defined on a discrete polarized domain (Σ , µ ) whosepolarization is given by some such function µ .2.1. Darboux transformations of discrete polarized curves.
We first defineDarboux transformations of discrete polarized curves.
Definition 2.1.
Two discrete polarized curves x, ˆ x : (Σ , µ ) → C are called a Darboux pair with parameter ˆ µ if, on every edge ( ij ),(2.1) cr( x i , x j , ˆ x j , ˆ x i ) = x i − x j x j − ˆ x j ˆ x j − ˆ x i ˆ x i − x i = ˆ µµ , for some non-zero constant ˆ µ ∈ R \ { } . We call one of the curves a Darbouxtransform of the other.A Darboux transformation is determined by the choice of the parameter ˆ µ and aninitial condition ˆ x i at some vertex i ∈ Σ.Recalling that a discrete curve x : (Σ , µ ) → C is arc-length polarized if | x i − x j | = µ ij on every edge ( ij ), we have the following proposition. Proposition 2.2.
Let x, ˆ x : (Σ , µ ) → C be a Darboux pair with parameter ˆ µ where x is arc-length polarized. Then ˆ x is also arc-length polarized if and only if | x i − ˆ x i | = µ at some vertex i ∈ Σ .Proof. Assume first that | x i − ˆ x i | = µ at some vertex i ∈ Σ. Then on an edge ( ij ),the definition of Darboux pair with parameter ˆ µ (2.1) tells usˆ x j − ˆ x i = ˆ µ (ˆ x i − x i )( x j − ˆ x i ) µ ij ( x i − x j ) + ˆ µ (ˆ x i − x i ) . Then a computation gives | µ ij ( x i − x j ) + ˆ µ (ˆ x i − x i ) | = µ ij ˆ µ | x j − ˆ x i | ;hence, | ˆ x j − ˆ x i | = ˆ µ | x j − ˆ x i | µ ij ˆ µ | x j − ˆ x i | = 1 µ ij . We can prove the converse claim similarly, by switching the roles of ˆ x j and x i . (cid:3) Permutability.
We now discuss the permutability between infinitesimal Dar-boux transformation and Darboux transformation of a discrete polarized curve. Let I be an interval in the reals, and recall from [3, Definition 2.7], that given a discretepolarized curve x , we have f : Σ × I → C is an infinitesimal Darboux transformationof x with parameter function m , if f i ( s ) = x i for some s ∈ I , and on every edge( ij ), f (cid:48) i f (cid:48) j ( f i − f j ) = µ ij m , ISCRETE MKDV EQUATION VIA DARBOUX TRANSFORMATION 3
Equivalently, we have that on every edge ( ij ), f i ( s ) , f j ( s ) : ( I, d s m ) → C are aDarboux pair of smooth polarized curves with parameter µ ij (see [2, Definition andLemma 2.2]).If ˆ x is a Darboux transform of x with parameter ˆ µ , then the permutability ofDarboux transformations of smooth polarized curves in [2, Theorem 2.8] tells usthat there exists an infinitesimal Darboux transformation ˆ f : Σ × I → C of ˆ x withparameter function m so that ˆ f i ( s ) = ˆ x i for some s , and for any fixed s ∈ I ,cr( f i ( s ) , f j ( s ) , ˆ f j ( s ) , ˆ f i ( s )) = ˆ µµ ij , that is, the two discrete curves f ( s ) and ˆ f ( s ) for any fixed s ∈ I are a Dar-boux pair with parameter ˆ µ . Thus, we have the permutability between Darbouxtransformation and infinitesimal Darboux transformation of a discrete polarizedcurve: Proposition 2.3.
Let f : Σ × I → C be an infinitesimal Darboux transformationof a discrete polarized curve x : (Σ , µ ) → C with parameter function m and let ˆ x bea Darboux transform of x with parameter ˆ µ . Then there is an infinitesimal Darbouxtransformation ˆ f of ˆ x with parameter function m so that, for any s ∈ I , ˆ f ( s ) is aDarboux transform of f ( s ) with parameter ˆ µ . Using this, we can also obtain the following result.
Proposition 2.4.
Let f i : ( I, d s m ) → C be a polarized curve with Darboux trans-forms f j and ˆ f i with parameters µ ij and ˆ µ , respectively, and suppose that ˆ f j is thesimultaneous Darboux transform of f j and ˆ f i with parameters ˆ µ and µ ij , respectively.If f i , f j , and ˆ f i are arc-length polarized, then ˆ f j is also arc-length polarized.Proof. Assuming f i , f j , and ˆ f i are arc-length polarized, then by [3, Proposition 2.6],we have that | f i − f j | = µ ij and | ˆ f i − f i | = µ for any s ∈ I . Fixing any s ∈ I ,we have that the two discrete curves f ( s ) and ˆ f ( s ) are a Darboux pair, where f ( s ) is discrete arc-length polarized. Hence, applying Proposition 2.2 tells us thatˆ f ( s ) is also discrete arc-length polarized, i.e., | ˆ f i − ˆ f j | = µ ij . Finally, applying[3, Proposition 2.6] again tells us that ˆ f j is arc-length polarized. (cid:3) Discrete potential mKdV equation
In this section, we use the aforementioned permutability of Darboux transformationsin Propositions 2.3 and 2.4 to introduce an efficient route to obtaining the discretepotential mKdV equation, using the well-known permutability theorems (cf. [1, 8]).To do this, consider the discrete motion x : Σ × ˜Σ → C , denoted by x ( n, k ) = x kn ,of a discrete planar curve x k defined as in [7, § n, k ) ∈ Σ × ˜Σ, wehave that(3.1) | x kn +1 − x kn | =: a n and | x k +1 n − x kn | =: b k are constant in k and n , respectively. JOSEPH CHO, WAYNE ROSSMAN, AND TOMOYA SENO
Without loss of generality, assume x kn = 0, x kn +1 = a n and x k +1 n = b k e iθ for some θ ∈ R . Excluding the solution a n + b k e iθ obtained via translation, x k +1 n +1 is uniquelydetermined with cross-ratios satisfyingcr( x kn , x kn +1 , x k +1 n +1 , x k +1 n ) = a n b k . Hence, using Proposition 2.2 we have the following result:
Proposition 3.1.
Let x : Σ × ˜Σ → C be a discrete motion of a discrete planarcurve x k : Σ → C such that (3.1) holds. Giving x k the arc-length polarization sothat µ ( n,n +1) = a n , we have that x k , x k +1 : (Σ , µ ) → C are a Darboux pair withparameter b k keeping arc-length polarization, for any k ∈ ˜Σ . Using Proposition 3.1, we can view the discrete motion x : Σ × ˜Σ → C as the imageof successive Darboux transformations keeping the arc-length polarization. ThenPropositions 2.3 and 2.4 tell us that there exists smooth arc-length polarized curves f : Σ × ˜Σ × I → C so that f kn , f kn +1 : ( I, d s m ) → C are Darboux pairs with parameter a n while f kn , f k +1 n : ( I, d s m ) → C are Darboux pairs with parameter b k .Therefore, the tangential angles θ kn , θ kn +1 , θ k +1 n and θ k +1 n +1 of f kn , f kn +1 , f k +1 n and f k +1 n +1 ,respectively, satisfy the semi-discrete potential mKdV equations via [3, Theorem3.2]: (cid:18) θ kn + θ kn +1 (cid:19) (cid:48) = a n sin (cid:18) θ kn +1 − θ kn (cid:19) , (cid:18) θ k +1 n + θ kn (cid:19) (cid:48) = b k sin (cid:18) θ k +1 n − θ kn (cid:19) , (cid:18) θ k +1 n +1 + θ k +1 n (cid:19) (cid:48) = a n sin (cid:18) θ k +1 n +1 − θ k +1 n (cid:19) , (cid:18) θ k +1 n +1 + θ kn +1 (cid:19) (cid:48) = b k sin (cid:18) θ k +1 n +1 − θ kn +1 (cid:19) . These equations are well-known partial differential equations that define B¨acklundtransformations of the smooth potential mKdV equation as seen in [8, Equations(7), (8)]. Using these equations, permutability of the transformation was obtainedin [8, Equation (9)] (see also [1]):tan (cid:32) θ k +1 n +1 − θ kn (cid:33) = b k + a n b k − a n tan (cid:32) θ k +1 n − θ kn +1 (cid:33) , which is the discrete potential mKdV equation. Summarizing, we have: Theorem 3.2.
The infinitesimal Bianchi cubes of Darboux transformations keepingarc-length polarization in both the smooth and discrete directions yields a 1-parameterfamily of solutions of the discrete mKdV equation.
Acknowledgements.
The authors gratefully acknowledge the support from theJSPS/FWF Bilateral Joint Project I3809-N32 “Geometric shape generation” andJSPS Grant-in-Aids for: JSPS Fellows 19J10679, Scientific Research (C) 15K04845,(C) 20K03585 and (S) 17H06127 (P.I.: M.-H. Saito).
ISCRETE MKDV EQUATION VIA DARBOUX TRANSFORMATION 5
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Email address : [email protected] (Wayne Rossman) Department of Mathematics, Graduate School of Science, Kobe Uni-versity, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
Email address : [email protected] (Tomoya Seno) Department of Mathematics, Graduate School of Science, Kobe Univer-sity, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
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