# Lax representation does not mean complete integrability

To browse Academia. Skip to main content. Log In Sign Up. Download Free PDF. Lax representation and complete integrability for the periodic relativistic Toda lattice Physics Letters A, Mario Bruschi. Lax representation and complete integrability for the periodic relativistic Toda lattice. October ; accepted for publication 23 November Communicated by A. Fordy Three different Lax representations for the periodic relativistic Toda lattice are exhibited.

The complete integrability of the system is also proven. Introduction We present three possible Lax representations for the periodic relativistic Toda lattice. The first one is the most natural extension ofthe one holding in the finite non-periodic case [1,2]. The second one seems the most convenient to handle the infinite case, as well as its non-abelian and two-dimensional generalizations. By direct computation, one can prove the following proposition: Proposition 1.

M are the Lax matrices for the non-periodic case, given in ref. Hence, all its N nontrivial coefficients are also in- tegrals of motion clearly functionally independent by inspection. To investigate the infinite case, we focused attention on the linear problem 1 Oafor which the direct and inverse problem is now solved ;work is in progress also on a two-dimensional extension. However, to prove the involutivity of the first integrals, it is convenient to use a further Lax representation of the dynamical system, which will be exhibited in the next section.

Symmetric Lax representation To derive a symmetric Lax representation, we first observe that the compatibility condition between lOa and lObwhich yields the equations of motion 9does not depend at all on the boundary conditions im- posed on the eigenfunction while, of course, the NXNmatrix representation of 10 does depend on them!

To show the complete integrability of the periodic relativistic Toda lattice it is thus enough to prove that any two zeros of det[L k ], among which there are certainly N functionally independent, are in involution.

This will be done in the following, with the help of a simple lemma, that can be proven by direct computation: Lemma 1. References  S. Ruijsenaars, Relativistic Toda system, to be published. Bruschi and 0. Ragnisco, Phys. A Zakharov, The inverse scattering method, in: Solitons, eds.

Bullough and P. Caudrey Springer, Berlin, Ragnisco, Direct and inverse problem for the infinite relativistic Toda lattice, in preparation. Flaschka and D.We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found for two members of the family investigated.

For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair, we construct a Miura-type transformation relating the original equation to the Korteweg—de Vries KdV equation.

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In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second-order nonlinear evolution equation. Another interesting relation of 1. A summary of some related KdV-type equations and 1. Some connections among KdV-type equations.

The relation between KdV and mKdV and the Airy equation is formal, in the sense that the KdV and mKdV equations are reduced to the Airy equation if the nonlinear effects can be neglected.

The other relations apply solutions of the original equation the starting point of the arrow to the target equations at the end of the arrow.

In the transformations regarding equation 1. Therefore, solutions of the mKdV equation can be transformed into solutions of the KdV equation and 1. The transformation relating equation 1. In spite of being nonlinear, homogeneity of equation 1. Sometimes, it will be more convenient to write the wavenumber as a function of cthat is. Solutions 1. The reader might argue that one could rescale 1. Actually, this work is motivated by some observations made by Sen et al. At the very beginning of p.

By SIdV, Sen et al. We, however, do not use this name in our paper. These are among the simplest PT symmetric advecting velocities beyond KdV. These observations aroused our interest. Moreover, they were quite stimulating and quite challenging.An unexplicit criterion for integrability is proposed. Examples of gauge equivalent inhomogeneous nonlinear evolution equations are presented. It is shown that in the nonintegrable cases the M-operators in their Lax representations possess unremovable pole singularities lying on the spectrum of the L-operators.

This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Manakov S. New York: Consultants Bureau, Google Scholar. Ablowitz M. Calogero F. Amsterdam: North Holland, Faddeev L.

Hamiltonian Method in the Theory of Solitons. Berlin: Springer-Verlag, Arkadi'ev V. Khristov E. Kaup D. London, A Fokas A. Gerdjikov V. Melnikov V. Latifi A. A Claude C. Download references. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. Reprints and Permissions. Gerdjikov, V. Complete integrability, gauge equivalence and Lax representation of inhomogeneous nonlinear evolution equations.

Theor Math Phys 92, — Download citation. Received : 27 July Issue Date : September To browse Academia. Skip to main content. Log In Sign Up. Download Free PDF. Lax representation and complete integrability for the periodic relativistic Toda lattice Physics Letters A, Orlando Ragnisco. Lax representation and complete integrability for the periodic relativistic Toda lattice.

October ; accepted for publication 23 November Communicated by A. Fordy Three different Lax representations for the periodic relativistic Toda lattice are exhibited. The complete integrability of the system is also proven. Introduction We present three possible Lax representations for the periodic relativistic Toda lattice. The first one is the most natural extension ofthe one holding in the finite non-periodic case [1,2]. The second one seems the most convenient to handle the infinite case, as well as its non-abelian and two-dimensional generalizations.

By direct computation, one can prove the following proposition: Proposition 1. M are the Lax matrices for the non-periodic case, given in ref. Hence, all its N nontrivial coefficients are also in- tegrals of motion clearly functionally independent by inspection. To investigate the infinite case, we focused attention on the linear problem 1 Oafor which the direct and inverse problem is now solved ;work is in progress also on a two-dimensional extension. However, to prove the involutivity of the first integrals, it is convenient to use a further Lax representation of the dynamical system, which will be exhibited in the next section.

Symmetric Lax representation To derive a symmetric Lax representation, we first observe that the compatibility condition between lOa and lObwhich yields the equations of motion 9does not depend at all on the boundary conditions im- posed on the eigenfunction while, of course, the NXNmatrix representation of 10 does depend on them!

To show the complete integrability of the periodic relativistic Toda lattice it is thus enough to prove that any two zeros of det[L k ], among which there are certainly N functionally independent, are in involution. This will be done in the following, with the help of a simple lemma, that can be proven by direct computation: Lemma 1.

References  S. Ruijsenaars, Relativistic Toda system, to be published. Bruschi and 0. Ragnisco, Phys. A Zakharov, The inverse scattering method, in: Solitons, eds. Bullough and P. Caudrey Springer, Berlin, Ragnisco, Direct and inverse problem for the infinite relativistic Toda lattice, in preparation.

Flaschka and D. McLaughlin, Prog. Olshanetsky and A. Perelomov, Phys. Related Papers. Notes on Integrable Systems. By Dr. Computable Integrability. By Elena Kartashova. Bogoyavlensky-Volterra and Birkhoff integrable systems.In this paper, we discuss an interaction between complex geometry and integrable systems.

Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax representation of the equations of motion.

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These systems can be realized as straight line motions on a Jacobi variety of a so-called spectral curve. In Section 2, we study a Lie algebra theoretical method leading to integrable systems and we apply the method to several problems.

In Section 3, we discuss the concept of the algebraic complete integrability a. Algebraic integrability means that the system is completely integrable in the sens of the phase space being folited by tori, which in addition are real parts of a complex algebraic tori abelian varieties. The method is devoted to illustrate how to decide about the a.

Finally, in Section 4 we study an a. The manifold invariant by the complex flow is covering of abelian variety. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Adler and P. Adler, P. Google Scholar. Belokolos, A.

### A family of wave equations with some remarkable properties

Bobenko, V. Its, and V. Matveev, Algebro-Geometric approach to nonlinear integrable equations Springer-Verlag, Eilbeck, V. Enolskii, V. Kuznetsov, and A. Tsiganov, Linear r-matrix algebra for classical separable systemsJ. Flaschka, The Toda lattice IPhys. Grammaticos, B. Dorozzi, and A. Ramani, Integrability of hamiltonians with third and fourth-degree polynomial potentialsJ. Griffiths, Linearizing flows and a comological interpretation of Lax equationsAmer. Griffiths and J.

Harris, Principles of algebraic geometry Wiley-Interscience, Kozlov, Symmetries, topology and resonances in Hamiltonian mechanics, A series of modern surveys in mathematics Springer-Verlag, Paris, Ser. I85 Lesfari and A. Elachab, On the integrability of the generalized Yang-Mills systemApplicationes Mathematicae Warsaw 31 3 Elachab, A connection between geometry and dynamical systemsJ.

Manakov, Remarks on the integrals of the Euler equations of the n-dimensional heavy topFunc.Jan L. We study Lax triples i. We begin with a natural integrable deformation of the principal chiral model.

Then, we show that all deformations linear in the spectral parameter are trivial unless we admit Lax representations in a larger space. We present an explicit example of triply orthogonal systems with Lax representation in the group. Finally, the obtained results are interpreted in the context of the soliton surfaces approach.

We consider geometric problems associated with Lax triples, that is, Lax representations consisting of three linear equations with a spectral parameter. They can be interpreted as integrable deformations of problems corresponding to one of the involved Lax pairs.

The problem of finding integrable deformations of surfaces in is an interesting and nontrivial task. There are just few papers on that subject; see, for example, [ 1 — 3 ]. However, we point out that the central result of [ 1 ], namely, Theoremis wrong. This theorem claims that for any augmented system of Gauss-Mainardi-Codazzi equations there exists an explicit Lax representation with the spectral parameter: system 3.

Unfortunately, one can easily check that the spectral parameter can be easily eliminated from this system by performing simple algebraic calculations. In this paper we suggest another methodology. We start from Lax representations of prescribed form in order to obtain special cases of integrable systems. We checked two cases: Lax representations with three different simple poles an integrable deformation of the principal chiral model; see Section 2 and Lax representations linear in the spectral parameter.

All -valued Lax representations linear in turned out to be trivial; see Section 3. However, there exists a nontrivial Lax representation in a larger space for a special class of orthogonal nets in [ 4 ]; see also Section 4. Finally, in Section 5we shortly present more general context for studying integrable differential geometry: soliton surfaces approach [ 5 ] and Lie point symmetries for introducing the spectral parameter [ 67 ].

Throughout this paper we usually use the Lie group instead oftaking into account the isomorphism of corresponding Lie algebras:. We can assume, for instance, is double covering ofso all our results can be projected on when necessary. In Section 4 we make use of another isomorphism.

Actually, also. The principal chiral model is defined by the following system of two equations [ 8 — 10 ]: whereare elements of a Lie algebra. Equations 2 are equivalent to The chiral model has the Lax representation where is the corresponding Lie group. The chiral model is integrable in the sense of the theory of solitons. In particular, Darboux transformation and multisoliton solutions are known [ 10 ]. Denoting we can express and in terms of : Then the first equation of 3 is identically satisfied and system 3 reduces to a single equation for : By changing variables, we transform 6 into Solutions to this equation are harmonic maps from into provided thatare real [ 911 ].

We propose an extension of the principal chiral model which is derived from the following Lax representation: where we assumetakes values in a Lie groupand, belong to the corresponding Lie algebra.

In particular, we can take. This case yields an integrable deformation of surfaces in. Compatibility conditions for 8 read Denoting we have, similarly to the chiral model case, Then, compatibility conditions 9 reduce to three equations for one function :.

Proposition 1. Any solution of the chiral model 6 admits an extension unique up to translations in to a solution of the deformed chiral model 11 such that for some.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. MathOverflow is a question and answer site for professional mathematicians.

It only takes a minute to sign up. I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation:. Now, it is not too difficult to verify that this indeed satisfies the zero-curvature representation, but I'm trying to figure out why we cannot use the Lax pairs:.

These matrices clearly satisfy the zero-curvature representation, but for some reason none of the notes I've been reading use them. What is the reason that they are not a valid Lax pair for the KdV equation? One way to see this, is that you want the zero-curvature representation to be useful and tell you something you didn't know before.

Your representation has the problem of being singular, in the sense that the Lax matrices have zero determinant.

## Integrable systems and complex geometry

This very same argument applies to the zero curvature representation if you remember how to pass from it to the Lax representation via the monodromy matrix. In any case, even if you ignore for the time being this problem and try to go through inverse scattering, you very soon hit the same wall. In addition to the answer by issoloroap, the article Prolongation structures of nonlinear evolution equations by Allan Fordy here is the Mathscinet link and here is its first page on Google books explains why "good" zero-curvature representations should live in semi simple Lie algebras.

Another important point is that the spectral parameter should be essential, i. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Integrability - conditions of lax pairs Ask Question. Asked 5 years, 8 months ago. Active 1 year, 11 months ago. Viewed times.

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