Dear Rinku Mishra, for getting an useful answer to your question, you should clarify to greater degree the situation. That is, at least the precise citation of the paper: The meaning of the term may depend on the context, and also on the means that researches use in their studies. To point at a precise location of the statement in the respective paper (where you have seen that the secular term will vanish) whould have been yet more handy

sorry sir,I don't know how uploading had been missed...the paper was 'linear and non linear dust acoustic waves in an inhomogenous plasma' in non linear analysis for solving the modified kdv equ I got this term.

Note that "secular" means the term that has some singularity. That is, if you have some function depending on x and this function behave around x0 as 1/(x-x0), you says that your function is secular. In the paper that you have given the phrase about disappear of secular term means that some expression, permitting formally the dividing by zero in its content at some value of its variable, in reality is finite there. For instance, this is a case at x=0 for sin(x)/x. What is about the paper under discussion, it seems that the authors have qualified as equilibrium conditions their relations (38) and (39): It is these relations that appeared as conditions for their field function \phi had not taken value \infty at some point.

As a matter of fact, I am not experienced in derivation of equations of the type of KdV. More better understanding of the situation might have appearred had you repeated the derivation of their equation in accordance with recipes that the authors propagandise (In the parer and in the other papers of theirs). The other way - to find out anybody of them and to ask him to help with respective difficulties.

Thanks sir

When performing series solutions of equations like KdV, the simplest approach leads to solutions that diverge because of the "secular" terms, e.g.,

x(t) = cos(w t) + a t sin (w t)

Here, the second term is secular because it grows in amplitude with time.

The problem with KdV is that the frequency of the oscillation changes as the size of the nonlinear term is increased, and the secular terms are engaged in a kind of algebraic combat to track this, but they fail in finite time.

An alternative expansion technique, Lindstedt's Method (or more generally "multiple time scale analysis"), permits perturbation of the oscillator frequency as well as the amplitude. This leads to an elegant series expansion which converges for all time (by making the secular terms vanish), and in fact provides new information about the solution, namely the shift of the oscillator frequency with the nonlinearity strength.

Similar "secular" terms arise when driving a simple (lossless) harmonic oscillator at its resonant frequency; the amplitude increases without limit as a function of time.

The larger problem is the word "secular" which (unfortunately) has a variety of different meanings in mathematical physics.

A few years ago I published a paper:

Plasma Physics and Controlled Fusion Volume 54 Number 1

A S Richardson and J M Finn 2012 Plasma Phys. Control. Fusion 54 014004

Symplectic integrators with adaptive time steps

(available on ResearchGate)

with an appendix about secularities and parametric instabilities, and their relation to problems that can occur in symplectic integrators with adaptive time steps.

Attached are two screenshots of a part of the appendix.

It wouldn't take two screenshot files, so here is the first

Thanks Sir