The Peregrine soliton is a localized nonlinear structure predicted to exist over 25 years ago, but not so far experimentally observed in any physical system1. It is of fundamental significance because it is localized in both time and space, and because it defines the limit of a wide class of solutions to the nonlinear Schrödinger equation (NLSE). Here, we use an analytic description of NLSE bre...

We study propagation of dispersion-managed solitons in optical fibers which are modeled by the nonlinear Schrödinger equation with a periodic dispersion coefficient. When the dispersion variations are weak compared to the average dispersion, we develop perturbation series expansions and construct asymptotic solutions at the first and second orders of approximation. Due to a parametric resonance...

We rigorously study the long time dynamics of solitary wave solutions of the nonlinear Schrödinger equation in time-dependent external potentials. To set the stage, we first establish the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schrödinger equation. We then show that when the external potential varies slowly in space compared to the size of the soliton, th...

We prove that Mathieu’s N = 2 supersymmetric Korteweg–de Vries equations with a = 1 or 4 admit Hirota’s n-supersoliton solutions, whose nonlinear interaction does not produce any phase shifts. For initial profiles that can not be distinguished from a one-soliton solution at times t ≪ 0, we reveal the possibility of a spontaneous decay and, within a finite time, transformation into a solitonic s...

We study symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime of a single-mode fiber. Key characteristics of such multipeak solitons, such as the formation mechanism, propagation stability, and shape-changing collisions, are revealed in detail. Our results show that this multipeak (symmetric or asymmetric) mode could be regarded as a sing...

Under the well-known bilinear method of Hirota, specific expression for N-soliton solutions (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada(gCDGKS) equation in fluid mechanics is given. By defining a novel restrictive condition on solutions, resonant Y-type and X-type soliton are generated. new proposed constraint, combined with velocity resonance module method, mixed solitons l...

We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term. Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints. We derive and solve ODEs governing the soliton dynamics, first for continuous solutions, and then for more general shock wave solutions with discontinuities. We...

In this article, the Enhanced ( / )-expansion method has been projected to find the traveling wave solutions for nonlinear evolution equations(NLEEs) via the (2+1)-dimensional Burgers equation. The efficiency of this method for finding these exact solutions has been demonstrated with the help of symbolic computation software Maple. By this method we have obtained many new types of complexiton s...

We employ the idea of Hirota’s bilinear method, to obtain some new exact soliton solutions for high nonlinear form of Multiple soliton solutions of secondorder Benjamin-Ono equation. Multiple singular soliton solutions were obtained by this method. Moreover, multiple singular soliton solutions were also derived.

In this paper, the tanh and sine–cosine methods are used to construct exact periodic and soliton solutions of nonlinear evolution equations arising in mathematical physics. Many new families of exact travelling wave solutions of the generalized Hirota–Satsuma coupled KdV system, generalized-Zakharov equations and (2 + 1)-dimensional Broer–Kaup– Kupershmidt system are successfully obtained. The ...