Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation

It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock type singularities are presented.Comment: 19 page

Classical Poisson structures and r-matrices from constrained flows

We construct the classical Poisson structure and $r$-matrix for some finite dimensional integrable Hamiltonian systems obtained by constraining the flows of soliton equations in a certain way. This approach allows one to produce new kinds of classical, dynamical Yang-Baxter structures. To illustrate the method we present the $r$-matrices associated with the constrained flows of the Kaup-Newell, KdV, AKNS, WKI and TG hierarchies, all generated by a 2-dimensional eigenvalue problem. Some of the obtained $r$-matrices depend only on the spectral parameters, but others depend also on the dynamical variables. For consistency they have to obey a classical Yang-Baxter-type equation, possibly with dynamical extra terms.Comment: 16 pages in LaTe

Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy

We discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.Comment: 20 pages, LaTeX2e, report to NEEDS in Leeds (1998), to be published in Theor. Math. Phy

On Separation of Variables for Integrable Equations of Soliton Type

We propose a general scheme for separation of variables in the integrable Hamiltonian systems on orbits of the loop algebra $\mathfrak{sl}(2,\Complex)\times \mathcal{P}(\lambda,\lambda^{-1})$. In particular, we illustrate the scheme by application to modified Korteweg--de Vries (MKdV), sin(sinh)-Gordon, nonlinear Schr\"odinger, and Heisenberg magnetic equations.Comment: 22 page

Binary Nonlinearization of Lax pairs of Kaup-Newell Soliton Hierarchy

Kaup-Newell soliton hierarchy is derived from a kind of Lax pairs different from the original ones. Binary nonlinearization procedure corresponding to the Bargmann symmetry constraint is carried out for those Lax pairs. The proposed Lax pairs together with adjoint Lax pairs are constrained as a hierarchy of commutative, finite dimensional integrable Hamiltonian systems in the Liouville sense, which also provides us with new examples of finite dimensional integrable Hamiltonian systems. A sort of involutive solutions to the Kaup-Newell hierarchy are exhibited through the obtained finite dimensional integrable systems and the general involutive system engendered by binary nonlinearization is reduced to a specific involutive system generated by mono-nonlinearization.Comment: 15 pages, plain+ams tex, to be published in Il Nuovo Cimento

Periodic cluster mutations and related integrable maps

One of the remarkable properties of cluster algebras is that any cluster, obtained from a sequence of mutations from an initial cluster, can be written as a Laurent polynomial in the initial cluster (known as the 'Laurent phenomenon'). There are many nonlinear recurrences which exhibit the Laurent phenomenon and thus unexpectedly generate integer sequences. The mutation of a typical quiver will not generate a recurrence, but rather an erratic sequence of exchange relations. How do we 'design' a quiver which gives rise to a given recurrence? A key role is played by the concept of 'periodic cluster mutation', introduced in 2009. Each recurrence corresponds to a finite dimensional map. In the context of cluster mutations, these are called 'cluster maps'. What properties do cluster maps have? Are they integrable in some standard sense? In this review I describe how integrable maps arise in the context of cluster mutations. I first explain the concept of 'periodic cluster mutation', giving some classification results. I then give a review of what is meant by an integrable map and apply this to cluster maps. Two classes of integrable maps are related to interesting monodromy problems, which generate interesting Poisson algebras of functions, used to prove complete integrability and a linearization. A connection to the Hirota–Miwa equation is explained

STUDI DESKRIPTIF LEVEL BERPIKIR GEOMETRI VAN HIELE SISWA DI SMP NEGERI PERCONTOHAN DI LEMBANG

Geometri sekolah mempunyai peluang besar untuk dipahami oleh siswa dibandingkan dengan cabang ilmu matematika yang lainnya. Hal ini dikarenakan pengenalan konsep dasar geometri sudah dikenal oleh siswa sejak usia dini, seperti mengenal bangun-bangun geometri. Namun beberapa penelitian menunjukkan bahwa masih banyak siswa yang mengalami kesulitan dalam belajar geeometri, khususnya pada tingkat SMP. Oleh karena itu diperlukan penelitian terhadap level berpikir geometri siswa. Penelitian ini bertujuan untuk mengetahui: (1) level berpikir geometri siswa di SMP Negeri percontohan di Lembang, dan (2) menelaah apakah pembelajaran geometri yang berlangsung di sekolah menerapkan tahapan pembelajaran Van Hiele atau tidak. Metode dalam penelitian ini merupakan studi deskriptif dengan subjek penelitian adalah siswa kelas IX dari dua sekolah menengah pertama di Lembang. Instrumen dalam penelitian ini terdiri dari: (1) instrumen tes, yaitu tes level berpikir geometri Van Hiele pada materi bangun datar. Hasil dari tes ini dianalisis dengan kategori level berpikir sebagai berikut: level 0 adalah tahap pengenalan; level 1 adalah tahap analisis; level 2 adalah tahap pengurutan; level 3 adalah tahap deduksi formal; dan level 4 adalah tahap akurasi. (2) Instrumen non tes, yaitu berupa wawancara terhadap guru dan murid. Berdasarkan hasil penelitian diperoleh kesimpulan bahwa: (1) secara keseluruhan siswa SMP telah memasuki tahap berpikir geometri Van Hiele. Sebagian besar siswa berada pada tahap pengenalan (level 0) yaitu 81,16%, sedangkan sisanya telah memasuki tahap analisis (level 1) sebesar 17,39% dan tahap pengurutan (level 2) sebesar 1,45%. (2) Pembelajaran geometri di sekolah kurang memperhatikan tahapan pembelajaran geometri Van Hiele---------- Student has a big opportunity to understand geometry because the basic concept has early familiar, such as know the geometry’s objects. However, some of the research were show that many student difficult to learn geometry, specifically for junior high school. Because of that, it necessary to research about the geometry level thinking. The goal of the research are to know: (1) student geometry level thinking at the model of junior high school in Lembang, and (2) observe the lesson geometry at school by use the phase of Van Hiele geometry learning. The method is descriptive study with the subject are the student from IX class of two junior high school in Lembang. The instrument is: (1) test instrument, is Van Hiele geometry level test. The result will be analysis by categories of Van Hiele: level 0 is visualization; level 1 is analysis; level 2 is informal deduction; level 3 is deduction; and level 4 is rigor. (2) Non-test instrument, is interview to the teacher and student. Base of the research, the conclusion are: (1) by and large the student has include the Van Hiele geometry level. Student at level 0 is 81, 16%, at level 1 is 17,3% and at level 2 is 1,45%. (2) School did’nt use the phase of Van Hiele geometry learning