Sturm-Liouville systems with rational Weyl functions: Explicit formulas and applications

Gohberg, Israel; Kaashoek, M. A.; Sakhnovich, Alexander

doi:10.1007/bf01195588

Cited by 37 publications

(108 citation statements)

References 9 publications

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“…Some other explicit solutions to the KdV equation known in the literature include algebraic solitons [20][21][22], rational solutions [22,23], various singular solutions [24][25][26] such as positons and negatons, solutions [22] to the periodic and other KdV equations, solutions [27] that are not quite as explicit but expressed in terms of certain projection operators and various other solutions [28,29]. It is already known that some rational solutions can be obtained by letting the bound-state energies go to zero in the N-soliton solutions.…”

Section: Some Other Methods For the Kdv Equationmentioning

confidence: 99%

Exact solutions to the focusing nonlinear Schrödinger equation

Aktosun¹,

Demontis²,

Mee³

2007

Inverse Problems

164

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Certain explicit solutions to the Korteweg-de Vries equation in the first quadrant of the xt-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schrödinger equation. In the reflectionless case such solutions reduce to pure N-soliton solutions. An illustrative example is provided.

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Section: Some Other Methods For the Kdv Equationmentioning

confidence: 99%

Exact solutions to the focusing nonlinear Schrödinger equation

Aktosun¹,

Demontis²,

Mee³

2007

Inverse Problems

164

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“…Now, we proved the following general theorem. 30) and is equivalent [50,84] to the compatibility condition (2.45) of the auxiliary systems (3.16), where…”

Section: Explicit Solutions Of Nonlinear Equationsmentioning

confidence: 99%

“…Thus, GBDT is a convenient tool to construct wave functions and explicit solutions of the nonlinear wave equations as well as to solve various direct and inverse problems. GBDT and its applications were treated or included as important examples in the papers [22,23,48,55,56,57,58,59,61,63,64,66,67,68,70,71,72,73,75] (see also [28,29,30,31,32,33,37]). Here we consider self-adjoint and skew-self-adjoint Dirac-type systems including the singular case corresponding to soliton-positon interaction and solve direct and inverse problems.…”

Section: Introductionmentioning

confidence: 99%

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Sakhnovich

2010

Math. Model. Nat. Phenom.

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Abstract. A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N -wave equation are reviewed. New results on explicit construction of the wave functions for radial Dirac equation are obtained.

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“…Incidentally, the higher-order differential polynomials in Q(x) just alluded to represent the Korteweg-deVries (KdV) invariants (i.e., densities associated with KdV conservation laws) and hence open the link to infinite-dimensional completely integrable systems (cf. [3], [16], [27], [28], [29], [30], [34], [45], [70], [71], [72], [75], [83], [84] and the references therein).…”

Section: Introductionmentioning

confidence: 99%