Analytic-Bilinear Approach to Integrable Hierarchies

1. Front Matter

   Pages i-xii

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2. Introduction

   • L. V. Bogdanov

   Pages 1-17

3. Hirota Bilinear Identity for the Cauchy Kernel

   • L. V. Bogdanov

   Pages 18-52

4. Rational Loops and Integrable Discrete Equations. I: Zero Local Indices

   • L. V. Bogdanov

   Pages 53-95

5. Rational Loops and Integrable Discrete Equations. II: Two-Component Case

   • L. V. Bogdanov

   Pages 96-121

6. Rational Loops and Integrable Discrete Equations. III: The General Case

   • L. V. Bogdanov

   Pages 122-155

7. Generalized KP Hierarchy

   • L. V. Bogdanov

   Pages 156-194

8. Multicomponent KP Hierarchy

   • L. V. Bogdanov

   Pages 195-233

9. On the ¯∂-Dressing Method

   • L. V. Bogdanov

   Pages 234-262

10. Back Matter

    Pages 263-264

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The subject of this book is the hierarchies of integrable equations connected with the one-component and multi component loop groups. There are many publications on this subject, and it is rather well defined. Thus, the author would like t.o explain why he has taken the risk of revisiting the subject. The Sato Grassmannian approach, and other approaches standard in this context, reveal deep mathematical structures in the base of the integrable hi­ erarchies. These approaches concentrate mostly on the algebraic picture, and they use a language suitable for applications to quantum field theory. Another well-known approach, the a-dressing method, developed by S. V. Manakov and V.E. Zakharov, is oriented mostly to particular systems and ex­ act classes of their solutions. There is more emphasis on analytic properties, and the technique is connected with standard complex analysis. The language of the a-dressing method is suitable for applications to integrable nonlinear PDEs, integrable nonlinear discrete equations, and, as recently discovered, for t.he applications of integrable systems to continuous and discret.e geometry. The primary motivation of the author was to formalize the approach to int.e­ grable hierarchies that was developed in the context of the a-dressing method, preserving the analytic struetures characteristic for this method, but omitting the peculiarit.ies of the construetive scheme. And it was desirable to find a start.­

• Complex analysis
• functional equation
• mathematical physics
• partial differential equation
• topological group
• partial differential equations

• L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, Russia

  L. V. Bogdanov

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