We obtain novel solutions of a coupled $\phi^4$, nonlinear Schr\"odinger (NLS) and modified Korteweg de Vries (MKdV) model which can be re-expressed as linear superposition either the sum or difference two hyperbolic kink pulse solutions. These results demonstrate that notion superposed extends to equations well.

Abstract We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense and remain close to multi-solitons. show that these necessarily pure For equation (KdV) modified (mKdV) particular, we obtain a characterization multi-solitons multi-breathers terms dispersion.

We propose a scheme for nonlinearizing linear equations to generate integrable systems of both the AKNS and the KN classes, based on the simple idea of dimensional analysis and detecting the building blocks of the Lax pair. Along with the well known equations we discover a novel integrable hierarchy of higher order nonholonomic deformations for the AKNS family, e.g. the KdV, the mKdV, the NLS a...

We study the traveling waves of the Nonlinear Schrödinger Equation in dimension one. Through various model cases, we show that for nonlinearities having the same qualitative behaviour as the standard Gross-Pitaevkii one, the traveling waves may have rather different properties. In particular, our examples exhibit multiplicity or nonexistence results, cusps (as for the Jones-Roberts curve in the...

We propose a new type of reduction for integrable systems of coupled matrix PDEs; this reduction equates one matrix variable with the transposition of another multiplied by an anti-symmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative mKdV equations and a new integrable variant of the massive Thirring model, in addition to already known systems. ...

The homotopy analysis method HAM is employed to obtain symbolic approximate solutions for nonlinear coupled equations with parameters derivative. These nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations. By choosing different values of the parameters in general formal numerical solutions, as a result, a v...

We expand a partial difference equation (P∆E) on multiple lattices and obtain the P∆E which governs its far field behaviour. The perturbative–reductive approach is here performed on well known nonlinear P∆Es, both integrable and non integrable. We study the cases of the lattice modified Korteweg–de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra–Kac–Van Moerbeke (VKVM) equa...