By M. J. Ablowitz, B. Prinari, A. D. Trubatch

ISBN-10: 0521534372

ISBN-13: 9780521534376

During the last thirty years major development has been made within the research of nonlinear waves--including "soliton equations", a category of nonlinear wave equations that come up often in such components as nonlinear optics, fluid dynamics, and statistical physics. The vast curiosity during this box may be traced to realizing "solitons" and the linked improvement of a style of resolution termed the inverse scattering rework (IST). The IST process applies to non-stop and discrete nonlinear Schrödinger (NLS) equations of scalar and vector kind. This paintings provides an in depth mathematical examine of the scattering idea, deals soliton options, and analyzes either scalar and vector soliton interactions. The authors offer complex scholars and researchers with an intensive and self-contained presentation of the IST as utilized to nonlinear Schrödinger structures.

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**Extra resources for Discrete and Continuous Nonlinear Schrödinger Systems **

**Sample text**

67) and consequently K (2) (x, y) ∓K (1) (x, y) K¯ (x, y) = ∗ . 66). Existence and uniqueness of solutions The question of existence and uniqueness of solutions of linear integral equations is usually examined by the use of the Fredholm alternative. 64) exist and are unique. 70) ¯ + y)ds = 0. 71) that vanishes identically for y < x. 71) by (h ∗1 , h ∗2 ), integrate in y, and use ∞ ∞ |h j (y)|2 dy = −∞ x |h j (y)|2 dy. One obtains ∞ −∞ |h 1 (y)|2 + |h 2 (y)|2 + ¯ + h 1 (s)h ∗2 (y) F(s ∞ −∞ h 2 (s)h ∗1 (y)F(s + y) + y) ds dy = 0.

The function φ(x, k j ) has solitons). If t → −∞, then x J x J . After passing through the J-th the form φ(x, k j ) ∼ e−ik j x (1, 0)T when x soliton, it will be of the form φ(x, k j ) ∼ a J (k j )e−ik j x (1, 0)T , where a J (k) is the transmission coefﬁcient relative to the J-th soliton. By repeating the argument, we ﬁnd e−ik j x 0 J φ(x, k j ) ∼ al (k j ) l= j+1 x x j+1 xj. Upon passing through the j-th soliton, since the corresponding state is a bound state, we get J φ(x, k j ) ∼ S j al (k j ) l= j+1 0 eik j x x j−1 .

51) If f ± (k) is analytic in the upper/lower k-plane and f ± (k) → 0 as |k| → ∞ for Im k ≷ 0, then P ± ( f ∓ )(k) = 0, P ± ( f ± )(k) = ± f ± (k). 52) which allows one, in principle, to ﬁnd m− (x, k). Note that, as |k| → ∞, m− (x, k) = I − 1 2πik +∞ −∞ m− (x, ξ )V(x, ξ )dξ + O(k −2 ). 54a) −2ikx ¯ (1) N (x, k)dk. 54b) Case of poles ¯ Suppose now that the potential is such that a(k) and a(k) have a ﬁnite number of simple zeros in the regions Im k > 0 and Im k < 0, respectively, which we J J¯ denote as k j , Im k j > 0 j=1 and k¯ j , Im k¯ j < 0 j=1 .

### Discrete and Continuous Nonlinear Schrödinger Systems by M. J. Ablowitz, B. Prinari, A. D. Trubatch

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