This is a write-up of the lectures on dispersionless equations in 1+1, 2+1 and 3+1 dimensions presented by one of us (RM) at "Eurasian International Center for Theoretical Physics" (EICTP). We provide pedagogical introduction to the subject and summarize well-known results and some recent developments in theory of integrable dispersionless equations. Almost all results presented are available in the literatures. We just add some new results related to dispersionless limits of integrable magnetic equations. On the contrary, some of the basic tools of the integrable dispersionless systems and related theoretical techniques are not available in a pedagogical format. For this reason we think that it seems worthwhile to present basics of theory of dispersionless equations here for the benefit of beginner students. We begin with a detailed exposition of the well-known dispersionless systems like dKdVE, dNLSE, dKPE, dDSE and other classical soliton equations. We present in detail some new dispersionless systems. Next we develop the dispersionless limits of known magnetic equations. Lastly, we discuss in full detail the Lax representation formulation of some presented dispersionless equations. On the basis of the material presented here one can proceed smoothly to read the recent developments in this field of integrable dispersionless equations and related topics.

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This article is about, the new analytical wave solutions of Kuralay-II equations (K-IIAE and K-IIBE) along new definition of derivative have been explored. For this purpose, expa function, extended sinh-Gordon equation expansion scheme and generalized Kudryashov schemes have been utilized. The resulting solutions are dark, bright, dark-bright, periodic, singular and other kinds of solitons. These results are obtained and also verified by Mathematica tool. Our gained solutions are newer than the present solutions in the literature. The gained results can also be fruitful for the development of model in future. The schemes used in this research are effective, easy and reliable to handle the other fractional non-linear partial differential equations (FNLPDEs).

This paper is a continuation of our previous work in which we studied a dispersionless limits of some integrable spin systems. Now, we shall present dispersionless limits of some integrable generalized Heisenberg ferromagnet equations

In the present paper, we study the integrable 2-layer generalized Heisenberg ferromagnet equation (HFE). The relation between this generalized HFE and differential geometry of curves is established. Using this relation we found the geometrical equivalent counterpart of the 2-layer spin system which is the 2-component KdV equation. Finally, the gauge equivalence between these equations is established.