Abundant Traveling Wave Structures of (1+1)-Dimensional Sawada–Kotera Equation: Few Cycle Solitons and Soliton Molecules

Traveling wave solutions have been well studied for various nonlinear systems.However,for high order nonlinear physical models,there still exist various open problems.Here,travelling wave solutions to the well-known fifth-order nonlinear physical model,the Sawada–Kotera equation,are revisited.Abundant travelling wave structures including soliton molecules,soliton lattice,kink-antikink molecules,peak-plateau soliton molecules,few-cycle-pulse solitons,double-peaked and triple-peaked solitons are unearthed.

Rogue Waves in Nonintegrable KdV-Type Systems

It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without modulation instability which is considered as the main formation mechanism of the rogue waves.

Novel nonlinear wave transitions and interactions for(2+1)-dimensional generalized fifth-order Kd V equation

The(2 + 1)-dimensional generalized fifth-order Kd V(2GKd V) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKd V equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped(XS/YS),asymmetric M-Shaped(MS), W-Shaped(WS), Space-Curved(SC) and Oscillation M-Shaped(OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.

Fermionization of Sharma-Tasso-Olver System

By applying the fermionization approach, the inverse version of the bosoniza- tion approach, to the Sharma-Tasso-Olver （STO） equation, three simple supersymmetric extensions of the STO equation are obtained from the Painlee analysis. Furthermore, some types of special exact solutions to the supersymmetric extensions are obtained.

Spectrum transformation and conservation laws of lattice potential KdV equation

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation whose solutions have nonzero backgrounds. The derivation is based on the fact that the scattering data a（z） is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schr5dinger spectral problem. The obtained conserved densities are asymptotic to zero when ｜n｜ （or ｜m｜） tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.

Equivalent and Nonequivalent Barotropic Modes for Rotating Stratified Flows

All the possible equivalent barotropic (EB) laminar solutions are firstly explored,and all the possible non-EB elliptic circulations and hyperbolic laminar modes of rotating stratified fluids are discovered in this paper.The EB circulations (including the vortex streets and hurricane like vortices) possess rich structures,because either the arbitrary solutions of arbitrary nonlinear Poisson equations can be used or an arbitrary two-dimensional stream function is revealed which may be broadly applied in atmospheric and oceanic dynamics,plasma physics,astrophysics and so on.The discovery of the non-EB modes disproves a known conjecture.

Recent Development in Symmetries and Integrability of Difference Equations

Difference equations or discrete systems are mathematical models of various fields such as physics, chemistry, biology, and economics and have been subjects of extensive study of both pure mathematicians and applied mathematicians. Through its interaction with modern integrable systems, the theory of difference equations is enriched greatly and has been undergoing a rapid development. SIDE-10, the tenth of a series of biennial conferences devoted to Symmetries and Integrability of Difference Equations and related topics, was held during 10-16 June, 2012 at Ningbo, China. It was sponsored and supported by the National Natural Science Foundation of China, Ningbo Association of Science and Technology, Ningbo University, Academy of Mathematics and Systems Science of Chinese Academy of Sciences, China University of Mining and Technology （Beijing）, Tsinghua University, and Shanghai University. The conference attracted over 100 participants from more than a dozen of countries. During the conference, 44 contributed talks were arranged and the topics covered by the meeting include