We study the relevance of various scalar equations, such as inviscid Burgers’, Korteweg-de Vries (KdV), extended KdV, and higher order equations (of Camassa-Holm type), as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified with two approaches. The first method consists in approximating the flow with two decoupled, cou...

In this paper we present an overview of the direct relation between differential invariants of projective type for curves in flat semisimple homegenous spaces and PDEs of KdV type. We describe the progress in the proof of a conjectured Theorem stating that for any such space there are geometric evolutions of curves that induce completely integrable evolutions on their invariants of projective-t...

We show that the uniform radius of spatial analyticity $$\sigma (t)$$ solution at time t for fifth order KdV–BBM equation cannot decay faster than 1/t large $$t>0$$ , given initial data is analytic with fixed _0$$ . This significantly improves a recent result by Carvajal and Panthee (On model, 2020. arXiv:2009.09328 ) where they established an exponential t.

Extended shallow water wave equations are derived, using the method of asymptotic expansions, from Euler (or wave) equations. These extended models valid one order beyond usual weakly nonlinear, long approximation, incorporating all appropriate dispersive and nonlinear terms. Specifically, first we derive Korteweg–de Vries (KdV) equation, then proceed with Benjamin–Bona–Mahony Camassa–Holm in (...

We show how the supersymmetric KdV equation can be obtained from the self–duality condition on Yang–Mills fields in four dimension associated with the graded Lie algebra OSp(2/1). We also obtain the hierarchy of Susy KdV equations from such a condition. We formulate the Susy KdV hierarchy as a vanishing curvature condition associated with the U(1) group and show how an Abelian self–duality cond...