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A direct method for studying soliton equations. (Ein direkter Ansatz zur Untersuchung von Solitonengleichungen.) (German) Zbl 0970.35121
The article provides a comprehensive survey of functional-analytic methods in the soliton theory. Also, it cannot be regarded as an introduction to the subject since many crucial concepts are assumed to be known.
To outline the main idea for the particular case of the KdV equation, let \(E\) be a Banach space, \(A\in{\mathcal L}(E)\) a continuous operator, \(B\in{\mathcal A}(E)\) an operator in a Banach quasi-ideal. Denoting \(L(x,t)= \exp(Ax+ A^3t)B\) and assuming the existence of the inverse \((1+ L)^{-1}\). Then the family of operators \(U= ((1+ L)^{-1}(AL+ LA))_x\), \(A(E)\) is a solution of the operator KdV equation \(U_t= U_{xxx}+ 3(UU_x+ U_xU)\). To restitute the scalar case, let \(\tau\) be a continuous trace on \(A(E)\), \(P\in{\mathcal L}(E)\) a projection of rank 1. Then, assuming \(UP= U\), \(U_xP= U_x\), the function \(u= \tau(U)\) resolves the common KdV equation.
The authors report on the traces on operator ideals, Marchenko and Hirota methods, the finite-dimensional cases (especially the negaton asymptotics), the diagonal operators (leading to infinite soliton superpositions), generalizations for the unbounded operators (by using the \(C_0\)-semigroups), and the Cauchy initial problem.
Several graphs of soliton superpositions and extensive literature conclude this instructive exposition.

35Q51 Soliton equations
47N20 Applications of operator theory to differential and integral equations
35Q53 KdV equations (Korteweg-de Vries equations)
47L20 Operator ideals