The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. Th... more The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by an example with 3 degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we prove that the closed (or periodic) n-particle Toda lattice can be framed in such a geometrical structure, and its well-known integrals of the motion can be obtained as spectral invariants of a "quasi-Nijenhuis recursion operator", that is, a tensor field N of type (1, 1) defined on the phase space of the lattice. This example and some of its generalizations are used to understand whether one can define in a reasonable sense a notion of involutive Poisson quasi-Nijenhuis manifold. A geometrical link between the open (or non periodic) and the closed Toda systems is also framed in the context of a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds.
The behavior of a class of solutions of the shallow water Airy system originating from initial da... more The behavior of a class of solutions of the shallow water Airy system originating from initial data with discontinuous derivatives is considered. Initial data are obtained by splicing together self-similar parabolae with a constant background state. These solutions are shown to develop velocity and surface gradient catastrophes in finite time and the inherent persistence of dry spots is shown to be terminated by the collapse of the parabolic core. All details of the evolution can be obtained in closed form until the collapse time, thanks to formation of simple waves that sandwich the evolving self-similar core. The continuation of solutions asymptotically for short times beyond the collapse is then investigated analytically, in its weak form, with an approach using stretched coordinates inspired by singular perturbation theory. This approach allows to follow the evolution after collapse by implementing a spectrally accurate numerical code, which is developed alongside a classical shock-capturing scheme for accuracy comparison. The codes are validated on special classes of initial data, in increasing order of complexity, to illustrate the evolution of the dry spot initial conditions on longer time scales past collapse.
We consider the geometrical aspects of the Krichever map in the context of Jacobian Super KP hier... more We consider the geometrical aspects of the Krichever map in the context of Jacobian Super KP hierarchy. We use the representation of the hierarchy based on the Faà di Bruno recursion relations, considered as the cocycle condition for the natural double complex associated with the deformations of super Krichever data. Our approach is based on the construction of the universal super divisor (of degree g), and a local universal family of geometric data which give the map into the Super Grassmannian.
We use ideas of the geometry of bihamiltonian manifolds, developed by Gel'fand and Zakharevich, t... more We use ideas of the geometry of bihamiltonian manifolds, developed by Gel'fand and Zakharevich, to study the KP equations. In this approach they have the form of local conservation laws, and can be traded for a system of ordinary differential equations of Riccati type, which we call the Central System. We show that the latter can be linearized by means of a Darboux covering, and we use this procedure as an alternative technique to construct rational solutions of the KP equations.
One of the cornerstones of the theory of integrable systems of KdV type has been the remark that ... more One of the cornerstones of the theory of integrable systems of KdV type has been the remark that the n-GD (Gel’fand–Dickey) equations are reductions of the full Kadomtsev–Petviashvilij (KP) theory. In this paper we address the analogous problem for the fractional KdV theories, by suggesting candidates of the “KP theories” lying behind them. These equations are called “KP(m) hierarchies,” and are obtained as reductions of a bigger dynamical system, which we call the “central system.” The procedure allowing passage from the central system to the KP(m) equations, and then to the fractional KdVnm equations, is discussed in detail in the paper. The case of KdV32 is given as a paradigmatic example.
We study a class of matrices with noncommutative entries, which were first considered by Yu. I. M... more We study a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endo-morphisms ” of a polynomial algebra. More explicitly their defining conditions read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: [Mij,Mkl] = [Mkj,Mil] (e.g. [M11,M22] = [M21,M12]). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we provide complete proofs that an inverse to a Manin matrix is again a Manin matrix and for the S...
Abstract. Gaudin algebras form a family of maximal commutative subalge-bras in the tensor product... more Abstract. Gaudin algebras form a family of maximal commutative subalge-bras in the tensor product of n copies of the universal enveloping algebra U(g) of a semisimple Lie algebra g. This family is parameterized by collections of pairwise distinct complex numbers z1,..., zn. We obtain some new com-mutative subalgebras in U(g)⊗n as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the Hamiltonians of bending flows and to the Gelfand–Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.
We observe that matrices with noncommutative elements such that: 1) elements in the same column c... more We observe that matrices with noncommutative elements such that: 1) elements in the same column commute 2) commutators of the cross terms are equal: [Mij, Mkl] = [Mkj, Mil] (e.g. [M11, M22] = [M21, M12]), behave almost as well as matrices with commutative elements. There is natural definition of the determinant, Cramer’s inversion formula (Manin), we prove the Cayley-Hamilton theorem, the Newton identities, facts about block matrices, etc. Such matrices are the simplest examples in Manin’s theory of ”noncommutative symmetries” (1987-90). Second, we demonstrate that such matrices are ubiquitous in quantum integrability. They enter Talalaev’s hep-th/0404153 breakthrough formulas: det(∂z − LGaudin(z)), det(1 − e−∂zTY angian(z)) for the ”quantum spectral curve”, appear in separation of variables and Capelli identities. It is known that Manin’s matrices (and their q-analogs) include ”RTT=TTR ” and Cartier-Foata matrices. Third, we show that linear algebra theorems established for such ma...
Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copie... more Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U(g) of a semisimple Lie algebra g. This family is parameterized by collections of pairwise distinct complex numbers z1,..., zn. We obtain some new commutative subalgebras in U(g) ⊗n as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand–Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.
We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser ... more We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n, R). The relation with the Lax formalism is also discussed.
The theory of three-layer density-stratified ideal fluids is examined with a view toward its gene... more The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamil...
Symmetry, Integrability and Geometry: Methods and Applications
Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinea... more Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrödinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.
The interplay between incompressibility and stratification can lead to non-conservation of horizo... more The interplay between incompressibility and stratification can lead to non-conservation of horizontal momentum in the dynamics of a stably stratified incompressible Euler fluid filling an infinite horizontal channel between rigid upper and lower plates. Lack of conservation occurs even though in this configuration only vertical external forces act on the system. This apparent paradox was seemingly first noticed by Benjamin (J. Fluid Mech., vol. 165, 1986, pp. 445–474) in his classification of the invariants by symmetry groups with the Hamiltonian structure of the Euler equations in two-dimensional settings, but it appears to have been largely ignored since. By working directly with the motion equations, the paradox is shown here to be a consequence of the rigid lid constraint coupling through incompressibility with the infinite inertia of the far ends of the channel, assumed to be at rest in hydrostatic equilibrium. Accordingly, when inertia is removed by eliminating the stratificat...
We study a class of matrices with noncommutative entries, which were first considered by Yu. I. M... more We study a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endomorphisms ” of a polynomial algebra. More explicitly their defining conditions read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: [Mij, Mkl] = [Mkj, Mil] (e.g. [M11, M22] = [M21, M12]). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we present the formulation of Manin matrices in terms of matrix (Leningrad) notations; provide ...
We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmann... more We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmannian.
Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the p... more Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single layer homogeneous fluid with a constant pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These demonstrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results.
The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is us... more The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite 2D channel. The Hamiltonian structure of the averaged equations is obtained directly from that of the Euler equations through the process of Hamiltonian reduction. Long-wave asymptotics together with the Boussinesq approximation of neglecting the fluids' inertia is then applied to reduce the leading order vertically averaged equations to the shallow-water Airy system, and thence, in a non-trivial way, to the dispersionless non-linear Schrödinger equation. The full non-Boussinesq system for the dispersionless limit can then be viewed as a deformation of this well known equation. In a perturbative study of this deformation, it is shown that at first order the deformed system possesses an infinite sequence of constants of the motion, thus casting this system within the framework of completely integrable equations. The Riemann invariants of the deformed model are then constructed, and some local solutions found by hodograph-like formulae for completely integrable systems are obtained.
For a stratified incompressible Euler fluid under gravity confined by rigid boundaries, sources o... more For a stratified incompressible Euler fluid under gravity confined by rigid boundaries, sources of vorticity are classified with the aim of isolating those which are sensitive to the topological configurations of density isopycnals, for both layered and continuous density variations. The simplest case of a two-layer fluid is studied first. This shows explicitly that topological sources of vorticity are present whenever the interface intersects horizontal boundaries. Accordingly, the topological separation of the fluid domain due to the interface–boundary intersections can contribute additional terms to the vorticity balance equation. This phenomenon is reminiscent of Klein’s ‘Kaffeelöffel’ thought-experiment for a homogeneous fluid (Klein,Z. Math. Phys., vol. 59, 1910, pp. 259–262), and it is essentially independent of the vorticity generation induced by the baroclinic term in the bulk of the fluid. In fact, the two-layer case is generalized to show that for the continuously stratif...
The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. Th... more The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by an example with 3 degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we prove that the closed (or periodic) n-particle Toda lattice can be framed in such a geometrical structure, and its well-known integrals of the motion can be obtained as spectral invariants of a "quasi-Nijenhuis recursion operator", that is, a tensor field N of type (1, 1) defined on the phase space of the lattice. This example and some of its generalizations are used to understand whether one can define in a reasonable sense a notion of involutive Poisson quasi-Nijenhuis manifold. A geometrical link between the open (or non periodic) and the closed Toda systems is also framed in the context of a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds.
The behavior of a class of solutions of the shallow water Airy system originating from initial da... more The behavior of a class of solutions of the shallow water Airy system originating from initial data with discontinuous derivatives is considered. Initial data are obtained by splicing together self-similar parabolae with a constant background state. These solutions are shown to develop velocity and surface gradient catastrophes in finite time and the inherent persistence of dry spots is shown to be terminated by the collapse of the parabolic core. All details of the evolution can be obtained in closed form until the collapse time, thanks to formation of simple waves that sandwich the evolving self-similar core. The continuation of solutions asymptotically for short times beyond the collapse is then investigated analytically, in its weak form, with an approach using stretched coordinates inspired by singular perturbation theory. This approach allows to follow the evolution after collapse by implementing a spectrally accurate numerical code, which is developed alongside a classical shock-capturing scheme for accuracy comparison. The codes are validated on special classes of initial data, in increasing order of complexity, to illustrate the evolution of the dry spot initial conditions on longer time scales past collapse.
We consider the geometrical aspects of the Krichever map in the context of Jacobian Super KP hier... more We consider the geometrical aspects of the Krichever map in the context of Jacobian Super KP hierarchy. We use the representation of the hierarchy based on the Faà di Bruno recursion relations, considered as the cocycle condition for the natural double complex associated with the deformations of super Krichever data. Our approach is based on the construction of the universal super divisor (of degree g), and a local universal family of geometric data which give the map into the Super Grassmannian.
We use ideas of the geometry of bihamiltonian manifolds, developed by Gel'fand and Zakharevich, t... more We use ideas of the geometry of bihamiltonian manifolds, developed by Gel'fand and Zakharevich, to study the KP equations. In this approach they have the form of local conservation laws, and can be traded for a system of ordinary differential equations of Riccati type, which we call the Central System. We show that the latter can be linearized by means of a Darboux covering, and we use this procedure as an alternative technique to construct rational solutions of the KP equations.
One of the cornerstones of the theory of integrable systems of KdV type has been the remark that ... more One of the cornerstones of the theory of integrable systems of KdV type has been the remark that the n-GD (Gel’fand–Dickey) equations are reductions of the full Kadomtsev–Petviashvilij (KP) theory. In this paper we address the analogous problem for the fractional KdV theories, by suggesting candidates of the “KP theories” lying behind them. These equations are called “KP(m) hierarchies,” and are obtained as reductions of a bigger dynamical system, which we call the “central system.” The procedure allowing passage from the central system to the KP(m) equations, and then to the fractional KdVnm equations, is discussed in detail in the paper. The case of KdV32 is given as a paradigmatic example.
We study a class of matrices with noncommutative entries, which were first considered by Yu. I. M... more We study a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endo-morphisms ” of a polynomial algebra. More explicitly their defining conditions read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: [Mij,Mkl] = [Mkj,Mil] (e.g. [M11,M22] = [M21,M12]). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we provide complete proofs that an inverse to a Manin matrix is again a Manin matrix and for the S...
Abstract. Gaudin algebras form a family of maximal commutative subalge-bras in the tensor product... more Abstract. Gaudin algebras form a family of maximal commutative subalge-bras in the tensor product of n copies of the universal enveloping algebra U(g) of a semisimple Lie algebra g. This family is parameterized by collections of pairwise distinct complex numbers z1,..., zn. We obtain some new com-mutative subalgebras in U(g)⊗n as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the Hamiltonians of bending flows and to the Gelfand–Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.
We observe that matrices with noncommutative elements such that: 1) elements in the same column c... more We observe that matrices with noncommutative elements such that: 1) elements in the same column commute 2) commutators of the cross terms are equal: [Mij, Mkl] = [Mkj, Mil] (e.g. [M11, M22] = [M21, M12]), behave almost as well as matrices with commutative elements. There is natural definition of the determinant, Cramer’s inversion formula (Manin), we prove the Cayley-Hamilton theorem, the Newton identities, facts about block matrices, etc. Such matrices are the simplest examples in Manin’s theory of ”noncommutative symmetries” (1987-90). Second, we demonstrate that such matrices are ubiquitous in quantum integrability. They enter Talalaev’s hep-th/0404153 breakthrough formulas: det(∂z − LGaudin(z)), det(1 − e−∂zTY angian(z)) for the ”quantum spectral curve”, appear in separation of variables and Capelli identities. It is known that Manin’s matrices (and their q-analogs) include ”RTT=TTR ” and Cartier-Foata matrices. Third, we show that linear algebra theorems established for such ma...
Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copie... more Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U(g) of a semisimple Lie algebra g. This family is parameterized by collections of pairwise distinct complex numbers z1,..., zn. We obtain some new commutative subalgebras in U(g) ⊗n as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand–Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.
We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser ... more We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n, R). The relation with the Lax formalism is also discussed.
The theory of three-layer density-stratified ideal fluids is examined with a view toward its gene... more The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamil...
Symmetry, Integrability and Geometry: Methods and Applications
Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinea... more Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrödinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.
The interplay between incompressibility and stratification can lead to non-conservation of horizo... more The interplay between incompressibility and stratification can lead to non-conservation of horizontal momentum in the dynamics of a stably stratified incompressible Euler fluid filling an infinite horizontal channel between rigid upper and lower plates. Lack of conservation occurs even though in this configuration only vertical external forces act on the system. This apparent paradox was seemingly first noticed by Benjamin (J. Fluid Mech., vol. 165, 1986, pp. 445–474) in his classification of the invariants by symmetry groups with the Hamiltonian structure of the Euler equations in two-dimensional settings, but it appears to have been largely ignored since. By working directly with the motion equations, the paradox is shown here to be a consequence of the rigid lid constraint coupling through incompressibility with the infinite inertia of the far ends of the channel, assumed to be at rest in hydrostatic equilibrium. Accordingly, when inertia is removed by eliminating the stratificat...
We study a class of matrices with noncommutative entries, which were first considered by Yu. I. M... more We study a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endomorphisms ” of a polynomial algebra. More explicitly their defining conditions read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: [Mij, Mkl] = [Mkj, Mil] (e.g. [M11, M22] = [M21, M12]). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we present the formulation of Manin matrices in terms of matrix (Leningrad) notations; provide ...
We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmann... more We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmannian.
Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the p... more Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single layer homogeneous fluid with a constant pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These demonstrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results.
The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is us... more The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite 2D channel. The Hamiltonian structure of the averaged equations is obtained directly from that of the Euler equations through the process of Hamiltonian reduction. Long-wave asymptotics together with the Boussinesq approximation of neglecting the fluids' inertia is then applied to reduce the leading order vertically averaged equations to the shallow-water Airy system, and thence, in a non-trivial way, to the dispersionless non-linear Schrödinger equation. The full non-Boussinesq system for the dispersionless limit can then be viewed as a deformation of this well known equation. In a perturbative study of this deformation, it is shown that at first order the deformed system possesses an infinite sequence of constants of the motion, thus casting this system within the framework of completely integrable equations. The Riemann invariants of the deformed model are then constructed, and some local solutions found by hodograph-like formulae for completely integrable systems are obtained.
For a stratified incompressible Euler fluid under gravity confined by rigid boundaries, sources o... more For a stratified incompressible Euler fluid under gravity confined by rigid boundaries, sources of vorticity are classified with the aim of isolating those which are sensitive to the topological configurations of density isopycnals, for both layered and continuous density variations. The simplest case of a two-layer fluid is studied first. This shows explicitly that topological sources of vorticity are present whenever the interface intersects horizontal boundaries. Accordingly, the topological separation of the fluid domain due to the interface–boundary intersections can contribute additional terms to the vorticity balance equation. This phenomenon is reminiscent of Klein’s ‘Kaffeelöffel’ thought-experiment for a homogeneous fluid (Klein,Z. Math. Phys., vol. 59, 1910, pp. 259–262), and it is essentially independent of the vorticity generation induced by the baroclinic term in the bulk of the fluid. In fact, the two-layer case is generalized to show that for the continuously stratif...
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Papers by G. Falqui