ISBN: 9781402061387

Theemergenceofanewparadigminscienceo?ersvastperspectivesfor future investigations, as well as providing fresh insight into existing areas of kno- edge, discovering hitherto unknown relati… More...

Theemergenceofanewparadigminscienceo?ersvastperspectivesfor future investigations, as well as providing fresh insight into existing areas of kno- edge, discovering hitherto unknown relations between them. We can observe this kind of process in connection with the appearance of the concept of so- tons [465]. Understandingthefactthatnonlinearmodesareasfundamentalas linear ones, with the advent of a rigorousformalism making it possible to ?nd exact solutions of a wide class of physically important nonlinear equations, gave rise to “a revolution that has quietly transformed the realm of science over the past quarter century” [392]. The inverse spectral (or scattering) transform (IST) method serves as the mathematical background for the soliton theory. The development of the IST formalism a?ects many ?elds of mathematics, revealing on frequent - casions unexpected links between them. For example, the theory of surfaces 3 in R can be considered as a chapter of the theory of solitons [468]. The modern version of IST is based on the dressing method proposed by - kharov and Shabat, ?rst in terms of the factorization of integral operators on a line into a product of two Volterra integral operators [474] and then using the Riemann–Hilbert (RH) problem [475]. The most powerful version ¯ ¯ of the dressing method incorporates the ? problem formalism. The ? pr- lem was put forward by Beals and Coifman [39, 40] as a generalization of the RH problem and was applied to the study of ?rst-order one-dimensional spectral problems. New Textbooks>Hardcover>Science>Mathematics>Mathematics, Springer Netherlands Core >2 >T<

ISBN: 9781402061387

The monograph is devoted to the systematic presentation of the so called dressing method for solving differential equations (both linear and nonlinear) of mathematical physics. The essenc… More...

The monograph is devoted to the systematic presentation of the so called dressing method for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivial solutions of a given equation from (maybe trivial) solution of the same or related equation. The Moutard and Darboux transformations discovered in XIX century as applied to linear equations, the Bäcklund transformation in differential geometry of surfaces, the factorization method, the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for soliton equations and its extension in terms of the d-bar formalism comprise the main objects of the book. Throughout the text, a generally sufficient linear experience of readers is exploited, with a special attention to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions. Various linear equations of classical and quantum mechanics are solved by the Darboux and factorization methods. An extension of the classical Darboux transformations to nonlinear equations in 1+1 and 2+1 dimensions, as well as its factorization are discussed in detail. The applicability of the local and non-local Riemann-Hilbert problem-based approach and its generalization in terms of the d-bar method are illustrated on various nonlinear equations. New Textbooks>Hardcover>Science>Mathematics>Mathematics, Springer Netherlands Core >2 >T<

ISBN: 9781402061387

The monograph is devoted to the systematic presentation of the so called dressing method for solving differential equations (both linear and nonlinear) of mathematical physics. The essenc… More...

The monograph is devoted to the systematic presentation of the so called dressing method for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivial solutions of a given equation from (maybe trivial) solution of the same or related equation. The Moutard and Darboux transformations discovered in XIX century as applied to linear equations, the Bäcklund transformation in differential geometry of surfaces, the factorization method, the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for soliton equations and its extension in terms of the d-bar formalism comprise the main objects of the book. Throughout the text, a generally sufficient linear experience of readers is exploited, with a special attention to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions. Various linear equations of classical and quantum mechanics are solved by the Darboux and factorization methods. An extension of the classical Darboux transformations to nonlinear equations in 1+1 and 2+1 dimensions, as well as its factorization are discussed in detail. The applicability of the local and non-local Riemann-Hilbert problem-based approach and its generalization in terms of the d-bar method are illustrated on various nonlinear equations. Trade Books>Hardcover>Science>Mathematics>Mathematics, Springer Netherlands Core >2<

ISBN: 9781402061387

The monograph is devoted to the systematic presentation of the so called "dressing method" for solving differential equations (both linear and nonlinear) of mathematical physics. The esse… More...

The monograph is devoted to the systematic presentation of the so called "dressing method" for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivial solutions of a given equation from (maybe trivial) solution of the same or related equation. The Moutard and Darboux transformations discovered in XIX century as applied to linear equations, the Bäcklund transformation in differential geometry of surfaces, the factorization method, the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for soliton equations and its extension in terms of the d-bar formalism comprise the main objects of the book. Throughout the text, a generally sufficient "linear experience" of readers is exploited, with a special attention to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions. Various linear equations of classical and quantum mechanics are solved by the Darboux and factorization methods. An extension of the classical Darboux transformations to nonlinear equations in 1+1 and 2+1 dimensions, as well as its factorization are discussed in detail. The applicability of the local and non-local Riemann-Hilbert problem-based approach and its generalization in terms of the d-bar method are illustrated on various nonlinear equations. Media > Books, Springer Netherlands<