A very natural construction of integrable extensions of soliton systems is presented. The extension is made on the level of evolution equations by a modification of the algebra of dynamical fields. The paper is motivated by recent works of Wen-Xiu Ma et al. (Comp. Math. Appl. 60 (2010) 2601, Appl. Math. Comp. 217 (2011) 7238), where new class of soliton systems, being nonlinear integrable couplings, was introduced. The general form of solutions of the considered class of coupled systems is described. Moreover, the decoupling procedure is derived, which is also applicable to several other coupling systems from the literature.
A systematic construction of a class of integrable hierarchy is discussed in terms of the twisted affine $A_{2r}^{(2)}$ Lie algebra. The zero curvature representation of the time evolution equations are shown to be classified according to its algebraic structure and according to its vacuum solutions. It is shown that a class of models admit both zero and constant (non zero) vacuum solutions. Another, consists essentially of integral non-local equations and can be classified into two sub-classes, one admitting zero vacuum and another of constant, non zero vacuum solutions. The two dimensional g…
We study numerically the integrable turbulence in the framework of the one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type using a new approach called the "growing of turbulence". In this approach, we add a small linear pumping term to the equation and start evolution from statistically homogeneous Gaussian noise. After reaching a certain level of average intensity, we switch off the pumping and examine the resulting integrable turbulence. For sufficiently small initial noise and pumping coefficient, and also for not very wide simulation box (basin length), we observe…
In this work, by using the Hirota bilinear method, we obtain one- and two-soliton solutions of integrable $(2+1)$-dimensional $3$-component Maccari system which is used as a model describing isolated waves localized in a very small part of space and related to very well-known systems like nonlinear Schrödinger, Fokas, and long wave resonance systems. We represent all local and Ablowitz-Musslimani type nonlocal reductions of this system and obtain new integrable systems. By the help of reduction formulas and soliton solutions of the $3$-component Maccari system, we obtain one- and two-soliton s…
New exact multi line soliton and periodic solutions with constant asymptotic values at infinity of the NVN integrable nonlinear evolution equation via dibar-dressing method
Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize f…
In this letter, we construct new meshy soliton structures by using two concrete (2+1)-dimensional integrable systems. The explicit expressions based on corresponding Cole-Hopf type transformations are obtained. Constraint equation ft+\sum_{j=1}^{N} h_j(y)f_{jx} = 0 shows that these meshy soliton structures can be linear or parabolic. Interaction between meshy soliton structure and Lump structure are also revealed.
In this letter, the unique exact lump and topological soliton solutions of integrable (2+1) dimensional Kundu Mukherjee Naskar (KMN) equation are obtained. These solutions have an unusual property that they can get curved in the plane arbitrarily due to the presence of an arbitrary function of space(x) and time(t) in their analytic forms. Due to this special feature, the solutions can be used to model the bending of optical solitonic beam, different types of wave structures in real physical experimental conditions. This novel feature, which is a rare property for a constant coefficient complet…
We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotka-Volterra, Toda lattice, and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional Toda lattice equations. Each of the delay-difference and delay-differential equations has the N-soliton solution, which depends on the delay parameter and converges to an N-soliton solution of a known soliton equation as the delay parameter ap…
Integrable curl-force Hamiltonians: bi-Hamiltonian structure, separability, and periodic orbits
Advanced Institute for Materials Research, Tohoku University（东北大学物质材料研究高等研究院）；Department of Mathematics, City St George’s, University of London（伦敦大学城市圣乔治学院数学系）
In this paper, based on the nonlinear fractional equations proposed by Ablowitz, Been, and Carr in the sense of Riesz fractional derivative, we explore the fractional coupled Hirota equation and give its explicit form. Unlike the previous nonlinear fractional equations, this type of nonlinear fractional equation is integrable. Therefore, we obtain the fractional $n$-soliton solutions of the fractional coupled Hirota equation by inverse scattering transformation in the reflectionless case. In particular, we analyze the one- and two-soliton solutions of the fractional coupled Hirota equation and…
