Integrability in Action: Solitons, Instability and Rogue Waves

Degasperis, A.; Lombardo, Sara

doi:10.1007/978-3-319-39214-1_2

Cited by 23 publications

(18 citation statements)

References 92 publications

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“…Among these basic solutions, we mention the Peregrine soliton [74], rationally localized in x and t over the background (2), the so-called Kuznetsov [62] -Kawata -Inoue [49] -Ma [66] soliton, exponentially localized in space over the background and periodic in time, the solution found by Akhmediev, Eleonskii and Kulagin in [7], periodic in x and exponentially localized in time over the background (2), known in the literature as the Akhmediev breather (AB), its elliptic generalizations [9,8], and its multi-soliton generalizations [52]. Generalizations of these solutions to the case of integrable multicomponent NLS equations, characterized by a richer spectral theory, have also been found [11,25,26,27,28].…”

Section: Introductionmentioning

confidence: 89%

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

2020

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The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. Using the finite gap method, two of us (PGG and PMS) have recently solved, to leading order and in terms of elementary functions of the initial data, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number of unstable modes. In this paper, concentrating on the simplest case of a single unstable mode, we study the periodic Cauchy problem of the AWs for the NLS equation perturbed by a linear loss or gain term. Using the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic model describing quantitatively how the solution evolves, after a suitable transient, into slowly varying lower dimensional patterns (attractors) in the (x, t) plane, characterized by ∆X = L/2 in the case of loss, and by ∆X = 0 in the case of gain, where ∆X is the x-shift of the position of the AW during the recurrence, and L is the period. This process is described, to leading order, in terms of elementary functions of the initial data. Since dissipation can hardly be avoided in all natural phenomena involving AWs, and since a small dissipation induces O(1) effects on the periodic AW dynamics, generating the slowly varying loss/gain attractors analytically described in this paper, we expect that these attractors, together with their generalizations corresponding to more unstable modes, will play a basic role in the theory of periodic AWs in nature.

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Section: Introductionmentioning

confidence: 89%

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

2020

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“…Thus, they have attracted a lot of attention in the physics and nonlinear waves communities in recent years. Analytical expressions of rogue waves have been derived in a large number of integrable nonlinear wave equations, such as the nonlinear Schrödinger (NLS) equation [19][20][21][22][23][24][25][26][27][28], the derivative NLS equation [29,30], the three-wave interaction equation [31], the Davey-Stewartson equations [32,33], and many others [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. Rogue waves have also been observed in water tanks [54,55] and optical fibers [56][57][58].…”

Section: Introductionmentioning

confidence: 99%

General Rogue Waves in the Boussinesq Equation

Yang

2020

J. Phys. Soc. Jpn.

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We derive general rogue wave solutions of arbitrary orders in the Boussinesq equation by the bilinear Kadomtsev-Petviashvili (KP) reduction method. These rogue solutions are given as Gram determinants with 2N − 2 free irreducible real parameters, where N is the order of the rogue wave. Tuning these free parameters, rogue waves of various patterns are obtained, many of which have not been seen before. Compared to rogue waves in other integrable equations, a new feature of rogue waves in the Boussinesq equation is that the rogue wave of maximum amplitude at each order is generally asymmetric in space. On the technical aspect, our contribution to the bilinear KP-reduction method for rogue waves is a new judicious choice of differential operators in the procedure, which drastically simplifies the dimension reduction calculation as well as the analytical expressions of rogue wave solutions.

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“…Choosing α = 1, and the initial data q 1 (x, 0) = (1+0.001e π 3 i cos(x))e −2i , using the integrating-factor method [43], we show that the dynamics is consistent with AB solution [44] (Fig. 15).…”

Section: The Quantitative Relation Between Fundamental Ab or Rw And Mmentioning

confidence: 77%

Modulational instability and homoclinic orbit solutions in vector nonlinear Schrödinger equation

Ling

Zhao

2019

Communications in Nonlinear Science and Numerical Simulation

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Modulational instability has been used to explain the formation of breather and rogue waves qualitatively. In this paper, we show modulational instability can be used to explain the structure of them in a quantitative way. We develop a method to derive general forms for Akhmediev breather and rogue wave solutions in a N -component nonlinear Schrödinger equations. The existence condition for each pattern is clarified clearly. Moreover, the general multi-high-order rogue wave solutions and multi-Akhmediev breather solutions for N -component nonlinear Schrödinger equations are constructed. The results further deepen our understanding on the quantitative relations between modulational instability and homoclinic orbits solutions.

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Integrability in Action: Solitons, Instability and Rogue Waves

Cited by 23 publications

References 92 publications

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

General Rogue Waves in the Boussinesq Equation

Modulational instability and homoclinic orbit solutions in vector nonlinear Schrödinger equation

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