Approximate solutions and scaling transformations for quadratic solitons

Sukhorukov, Andrey A.

doi:10.1103/physreve.61.4530

Cited by 20 publications

(26 citation statements)

References 31 publications

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“…We consider a two-component model for quadratic solitons [20] Du á2u 0 =ia z +a t2 +u* v, and substitute in the above system, we obtain 2 0= d2 2 -U + UV,…”

Section: Two-component Model For Quadratic Solitonsmentioning

confidence: 99%

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Exact solutions of reaction-diffusion systems and nonlinear wave equations

Rodrigo

Mimura

2001

Japan J. Indust. Appl. Math.

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We propose a general method to find exact travelling and standing wave solutions of reaction-diffusion systems and nonlinear wave equations. The method is applied to several well-known reaction-diffusion systems such as a competition-diffusion system of LotkaVolterra type, the Gray-Scott model, a simplification of the Noyes-Field model for the Belousov-Zhabotinskii reaction, and two-and three-component models for quadratic solitons. We also find exact solutions of several generalized nonlinear dispersive equations occurring in mathematical physics.

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“…We consider a two-component model for quadratic solitons [20] Du á2u 0 =ia z +a t2 +u* v, and substitute in the above system, we obtain 2 0= d2 2 -U + UV,…”

Section: Two-component Model For Quadratic Solitonsmentioning

confidence: 99%

“…We consider a three-component model for quadratic solitons [20] 8u 82u 0=ia z +a dz +u* w, 2 0 = i j-+ ^ 2 -,ßl v+Xv*w, z 0 = ia ^z + z -ßw+ 2 (u 2 +v2 ), where i2 = -1, the constants v, ,Q, /ßl , and x are positive, and * denotes complex conjugate.…”

Section: Three-component Model For Quadratic Solitonsmentioning

confidence: 99%

Exact solutions of reaction-diffusion systems and nonlinear wave equations

Rodrigo

Mimura

2001

Japan J. Indust. Appl. Math.

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“…2. Finally, we note that the above approach can be extended to describe quadratic solitons in bulk media and also three-wave trapping under conditions of multi-step cascading [10,11] . The author is grateful to Yuri S. Kivshar for useful comments , and to Adrian Ankiewicz for a critical reading of this manuscript.…”

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confidence: 97%

Analytical Description of Quadratic Parametric Solitons

Sukhorukov

2001

Soliton-Driven Photonics

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Abstract. We study quadratic solitons supported by two-wave parametric interaction in X(2) nonlinear media. We obtain very accurate explicit solutions for bright solitons with the help of a specially developed analytical approach.Nowdays, the study of the physics of spatial optical solitons is an active area of research. In particular, parametric solitons, i.e. beams composed of mutually trapped fundamental and harmonic waves that propagate with undistorted profiles, were observed in media with quadratic (i.e. X(2)) nonlinearity, and their unique features can be utilized for the creation of alloptical information processing devices [1] . Many studies have been devoted to the theoretical analysis of quadratic solitons [2]. It was demonstrated that even in the simplest case of "type I" interaction between the fundamental frequency (FF) and second-harmonic (SH) waves in a planar waveguide, the soliton profiles depend on both the linear phase mismatch parameter, characterizing the nonlinear medium, and the propagation constant, related to the soliton wave number modified by nonlinear self-action. However, the governing equations are not integrable, and exact analytical expressions for the soliton profiles are known only for particular parameter values. If the parameters are close to these special values, approximate solutions can be obtained by asymptotic or variational techniques [3,4]. On the other hand, a variational approach, with Gaussian trial functions, can be used to find approximate soliton envelopes in the whole parameter range, but then the soliton tails (or far-field asymptotics) are not described corre ctly [3,5]. In this paper we introduce a different approach to find very accurate approximate solutions which (i) are valid for any values of the soliton parameters, and (ii) reduce to the exact solutions for the special cases.The parametric interaction between the FF and SH beams in a slab waveguide can be described by the following set of coupled equations [6] III A.D. Boardman and A.P. Sukhorukov (eds.) ,

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“…3.3). Kivshar, Sukhorukov, and Saltiel [1999]; Sukhorukov [2000] considered formation of solitons involving two FF components (A and B) that are coupled together through a single SH component (S). The soliton formation can be described by the following set of coupled equations that are obtained in the slowly varying envelope approximation, where σ AS and σ BS are proportional to the elements of the second-order susceptibility tensor, as discussed in the previous sections, ∆k AS = k S − 2k A and ∆k BS = k S − 2k B are the wave-vector mismatch parameters for the A-S and B-S parametric interaction processes, respectively.…”

Section: Two-color Parametric Solitonsmentioning

confidence: 99%

Multistep parametric processes in nonlinear optics

2005

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We present a comprehensive overview of different types of parametric interactions in nonlinear optics which are associated with simultaneous phase-matching of several optical processes in quadratic nonlinear media, the so-called multistep parametric interactions. We discuss a number of possibilities of double and multiple phase-matching in engineered structures with the sign-varying second-order nonlinear susceptibility, including (i) uniform and non-uniform quasi-phase-matched (QPM) periodic optical superlattices, (ii) phase-reversed and periodically chirped QPM structures, and (iii) uniform QPM structures in non-collinear geometry, including recently fabricated two-dimensional nonlinear quadratic photonic crystals. We also summarize the most important experimental results on the multi-frequency generation due to multistep parametric processes, and overview the physics and basic properties of multi-color optical parametric solitons generated by these parametric interactions.

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Approximate solutions and scaling transformations for quadratic solitons

Cited by 20 publications

References 31 publications

Exact solutions of reaction-diffusion systems and nonlinear wave equations

Exact solutions of reaction-diffusion systems and nonlinear wave equations

Analytical Description of Quadratic Parametric Solitons

Multistep parametric processes in nonlinear optics

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