This thesis is concerned about various aspects of 1+1 dimensional non- linear evolution equations. In particular, it is devoted to the study of those properties which can be obtained by the application of non-linear transformations, such as Cole-Hopf, Miura and reciprocal-type transformations. The concept of Backlund Chart is introduced to depict succinctly the rela- tionships interconnecting various non-linear systems. An account of the Painleve Test, an integrability test for non-linear partial differential equations, is given. Indeed, the conjecture asserting that such a test provides sufficient condi- tions for integrability has not yet been proved. However, the Painleve Analysis indicates how to relate to non-linear integrable systems equations, termed "singularity manifold" equations. The latter exhibit a particular structure which plays a key role in the present investigation. An overview on reciprocal-type transformations follows, especially in con- nection with their importance in linking non-linear evolution equations. Then, a Backlund Chart which comprises the Caudrey-Oodd-Gibbon (COG) and Kaup-Kupershmidt (KK) equations is constructed. It provides an explicit link between such equations and their respective singularity manifold equations. New hierarchies of integrable non-linear evolution equations are obtained via reciprocal-type transformations. They exhibit a novel invariance and have base member a Kawamoto-type equation. Furthermore, the spatial part of new auto-Backlund transformations, for both the COG and KK hierarchies are con- structed. The symmetry structure of a Kawamoto equation is subsequently stu- died. It provides the first example of a non-linear evolution equation exhibiting a very rich symmetry structure consisting of four hierarchies of (infinitesimal) symmetry generators. Remarkably, such a symmetry group is non-Abelian in contrast with all the systems formerly studied. The Hamiltonian and bi-Hamiltonian formulation of the Kawamoto equa- tion, as well as its hereditary recursion operator are obtained via the links in the Backlund Chart. Subsequently, an analog study is developed for non-linear systems related to the Korteweg-deVries (KdV) equation. An extensive Backlund Chart which, in addition to the well-established links between the KdV, modified KdV and the Harry Dym equation, incorporates the KdV singularity manifold equation and the KdV "zero soliton" equation is then constructed. It reveals the close anal- ogy between the Harry Dym and Kawamoto equations which turn out to have isomorphic symmetry groups. Finally, a new integrability test is proposed which has been termed the Expan8ion Te8t. It represents an extension of the Painleve test. Moreover, the interacting soliton structure is readily derived. Following an introduction to the concept of interacting solitons, the Expansion Test is presented together with significant subcases. The relation of the Test to the recursion operator is given. In particular, it is shown how the expansion ansatz of the new test can be retrieved from the related recursion operator. The close connection between the singularity ma.nifold equation and the interacting soliton structure follows naturally from the Expansion Test. Specifically, the singularity manifold and the "zercrsoliton" equations are shown to be linked by an involutive transforma- tion here termed the Soliton-Singularity Tran8/orm. This is connected to invariance under the Mobius group of transformations exhibited by singularity manifold equations. This, in turn, gives rise to a "Darboux-like" invariance for spectral properties of the recursion operator. An important implication of the ScrSi Transform is that the standard Painleve analysis necessarily requires that Q =-lorQ =-2. A new invariance is shown to connect explicit solutions of the celebrated Liouville equation to the Miura transform in a novel manner.

Invariance properties and symmetry structure of integrable systems %Z Thesis (Ph.D.)--University of Waterloo (Canada) / Carillo, Sandra. - (1989), pp. 1-155.

Invariance properties and symmetry structure of integrable systems %Z Thesis (Ph.D.)--University of Waterloo (Canada)

Carillo Sandra
Primo
1989

Abstract

This thesis is concerned about various aspects of 1+1 dimensional non- linear evolution equations. In particular, it is devoted to the study of those properties which can be obtained by the application of non-linear transformations, such as Cole-Hopf, Miura and reciprocal-type transformations. The concept of Backlund Chart is introduced to depict succinctly the rela- tionships interconnecting various non-linear systems. An account of the Painleve Test, an integrability test for non-linear partial differential equations, is given. Indeed, the conjecture asserting that such a test provides sufficient condi- tions for integrability has not yet been proved. However, the Painleve Analysis indicates how to relate to non-linear integrable systems equations, termed "singularity manifold" equations. The latter exhibit a particular structure which plays a key role in the present investigation. An overview on reciprocal-type transformations follows, especially in con- nection with their importance in linking non-linear evolution equations. Then, a Backlund Chart which comprises the Caudrey-Oodd-Gibbon (COG) and Kaup-Kupershmidt (KK) equations is constructed. It provides an explicit link between such equations and their respective singularity manifold equations. New hierarchies of integrable non-linear evolution equations are obtained via reciprocal-type transformations. They exhibit a novel invariance and have base member a Kawamoto-type equation. Furthermore, the spatial part of new auto-Backlund transformations, for both the COG and KK hierarchies are con- structed. The symmetry structure of a Kawamoto equation is subsequently stu- died. It provides the first example of a non-linear evolution equation exhibiting a very rich symmetry structure consisting of four hierarchies of (infinitesimal) symmetry generators. Remarkably, such a symmetry group is non-Abelian in contrast with all the systems formerly studied. The Hamiltonian and bi-Hamiltonian formulation of the Kawamoto equa- tion, as well as its hereditary recursion operator are obtained via the links in the Backlund Chart. Subsequently, an analog study is developed for non-linear systems related to the Korteweg-deVries (KdV) equation. An extensive Backlund Chart which, in addition to the well-established links between the KdV, modified KdV and the Harry Dym equation, incorporates the KdV singularity manifold equation and the KdV "zero soliton" equation is then constructed. It reveals the close anal- ogy between the Harry Dym and Kawamoto equations which turn out to have isomorphic symmetry groups. Finally, a new integrability test is proposed which has been termed the Expan8ion Te8t. It represents an extension of the Painleve test. Moreover, the interacting soliton structure is readily derived. Following an introduction to the concept of interacting solitons, the Expansion Test is presented together with significant subcases. The relation of the Test to the recursion operator is given. In particular, it is shown how the expansion ansatz of the new test can be retrieved from the related recursion operator. The close connection between the singularity ma.nifold equation and the interacting soliton structure follows naturally from the Expansion Test. Specifically, the singularity manifold and the "zercrsoliton" equations are shown to be linked by an involutive transforma- tion here termed the Soliton-Singularity Tran8/orm. This is connected to invariance under the Mobius group of transformations exhibited by singularity manifold equations. This, in turn, gives rise to a "Darboux-like" invariance for spectral properties of the recursion operator. An important implication of the ScrSi Transform is that the standard Painleve analysis necessarily requires that Q =-lorQ =-2. A new invariance is shown to connect explicit solutions of the celebrated Liouville equation to the Miura transform in a novel manner.
1989
0-315-49213-9
nonlinear evolution equation; recursion operator; spectral problems; Baecklund transformations
03 Monografia::03a Saggio, Trattato Scientifico
Invariance properties and symmetry structure of integrable systems %Z Thesis (Ph.D.)--University of Waterloo (Canada) / Carillo, Sandra. - (1989), pp. 1-155.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1678544
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