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Over the past fifteen years the theory of solitons and the related theory of integrable nonlinear evolution equations in two space-time dimensions has attracted a large number of research workers of different orientations ranging from algebraic geometry to applied hydrodynamics. Modern mathematical physics has witnessed the development of a vast new area of research devoted to this theory and called the inverse scattering method of solving nonlinear equations (other names are: the inverse spectral transform, the method of isospectral deformations and, more colloquially, the L-A pair method).

The dynamical system to be considered is generated by the nonlinear equation 1.1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaala % aabaGaeyOaIyRaeqiYdKhabaGaeyOaIyRaamiDaaaacqGH9aqpcqGH % sisldaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabeI8a5b % qaaiabgkGi2kaadIhadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSIa % aGOmaiaadIhadaabdaqaaiabeI8a5bGaay5bSlaawIa7amaaCaaale % qabaGaaGOmaaaakiabeI8a5baa!5051!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$i\frac{{\partial \psi }}{{\partial t}} = - \frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + 2x{\left| \psi \right|^2}\psi $$ with the initial condition 1.2 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdKNaai % ikaiaadIhacaGGSaGaamiDaiaacMcadaabbaqaamaaBaaaleaacaWG % 0bGaeyypa0JaaGimaaqabaGccqGH9aqpaiaawEa7aiabeI8a5jaacI % cacaWG4bGaaiykaiaac6caaaa!4620!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\psi (x,t)|_{t = 0} = \psi (x).$$ .

In Chapter I we analyzed the mapping % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaacQ % dacaGGOaGaeuiQdKLaaiikaiaadIhacaGGPaGaaiilaiqbfI6azzaa % caGaaiikaiaadIhacaGGPaGaaiykaiabgkziUkaacIcacaWGIbGaai % ikaiabeU7aSjaacMcacaGGSaGabmOyayaacaGaaiikaiabeU7aSjaa % cMcacaGG7aGaeq4UdW2aaSbaaSqaaiaadQgaaeqaaOGaaiilaiabeo % 7aNnaaBaaaleaacaWGQbaabeaakiaacMcaaaa!5458!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$F:(\Psi (x),\dot \Psi (x)) \to (b(\lambda ),\dot b(\lambda );\lambda _j ,\gamma _j )$$ from the functions Ψ (x), % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuiQdKLbai % aacaGGOaGaamiEaiaacMcaaaa!39E2!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\dot \Psi (x)$$ to the transition coefficients and discrete spectrum of the auxiliary linear problem. We saw that for both rapidly decreasing and finite density boundary conditions this “change of variables” makes the dynamics quite simple because the time evolution of the transition coefficients for the continuous and discrete spectra becomes linear.

In this chapter we return to the Hamiltonian formulation of the NS model in order to discuss the basic transformation of the inverse scattering method % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQ % dadaqadaqaaiabeI8a5naabmaabaGaamiEaaGaayjkaiaawMcaaiaa % cYcacuaHipqEgaqeamaabmaabaGaamiEaaGaayjkaiaawMcaaaGaay % jkaiaawMcaaiabgkziUoaabmaabaGaamOyamaabmaabaGaeq4UdWga % caGLOaGaayzkaaGaaiilaiqadkgagaqeamaabmaabaGaeq4UdWgaca % GLOaGaayzkaaGaai4oaiaaykW7cqaH7oaBdaWgaaWcbaGaamOAaaqa % baGccaGGSaGaeq4SdC2aaSbaaSqaaiaadQgaaeqaaaGccaGLOaGaay % zkaaaaaa!57BB!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$f:\left( {\psi \left( x \right),\bar \psi \left( x \right)} \right) \to \left( {b\left( \lambda \right),\bar b\left( \lambda \right);\,{\lambda _j},{\gamma _j}} \right)$$ from the Hamiltonian standpoint. We shall describe the Poisson structure on the scattering data of the auxiliary linear problem induced through f from the initial Poisson structure defined in Chapter I. Under the rapidly decreasing or finite density boundary conditions, the NS model proves to be a completely integrable system, with f defining a transformation to action-angle variables. In particular, we will show that the integrals of the motion introduced in Chapter I are in involution. In these terms scattering of solitons amounts to a simple canonical transformation.

In this chapter we shall give a list of typical examples and establish their general properties: the zero curvature representation and the Hamiltonian formulation. Then, motivated by these examples, we shall outline a general scheme for constructing integrable equations and their solutions based on the matrix Riemann problem. A detailed study of the most important models and the Hamiltonian interpretation of the general scheme will be presented in the following chapters. The examples to be considered fall into two classes: dynamical systems generated by partial differential evolution equations (continuous models), and evolution systems of difference type (lattice models).

We shall give a complete list of results pertaining to two fundamental continuous models, the HM and SG models. For the rapidly decreasing boundary conditions we shall analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and the discrete spectrum, and show how to solve the inverse problem, i. e. how to construct the mapping F ^−1. We shall see that these models allow an r-matrix approach, which will enable us to show that F is a canonical transformation to variables of action-angle type. It will thus be proved that the HM and SG models are completely integrable Hamiltonian systems. We shall also present a Hamiltonian interpretation of the change to light-cone coordinates in the SG model. To conclude this chapter, we shall explain that in some sense the LL model is the most universal integrable system with two-dimensional auxiliary space.

Here we shall give a complete list of results pertaining to the Toda model, a fundamental model on the lattice. We will show that the r-matrix approach applies to this case and may be used to prove the complete integrability of the model in the quasi-periodic case. For the rapidly decreasing boundary conditions we will analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and outline a method for solving the inverse problem, i.e. for F ^−1 constructing. On the basis of the r-matrix approach it will be shown that F is a canonical trans formation to action-angle type variables establishing the complete integrability of the Toda model in the rapidly decreasing case. We shall also define a lattice version of the LL model, the most general integrable lattice system with two-dimensional auxiliary space.

In this chapter we shall summarize and generalize our experience in describing integrable models gained from the study of particular examples. The principal entities of the inverse scattering method and its Hamiltonian interpretation were the auxiliary linear problem operator L = d/dx − U ( x, λ ) and the fundamental Poisson brackets for U ( x, λ ) involving the r-matrix. Similar objects were introduced for lattice models. We will show that these notions have a simple geometric interpretation.

This conclusion is intended for those who have read the book to the end. We hope that the main text and the notes to separate chapters have furnished a convincing evidence of how rich from the mathematical point of view, both conceptually and technically, is the subject of solitons and integrable partial differential equations. In fact, the inverse scattering method naturally intertwines various branches of mathematics: differential geometry, the theory of Lie groups and Lie algebras and their representations, complex and functional analysis. All of them serve one common purpose, to classify integrable equations and describe their solutions. As a result, the traditional parts of these branches, such as Hamiltonian formalism, affine Lie algebras, or the Riemann problem are seen in a new light.