A new integrable equation with cuspons and periodic cuspons

Abstract

In this paper, we derive a new integrable equation from the KdV equation and give its Lax pair. By applying the bifurcation method of dynamical systems, cuspons and periodic cuspons of the integrable equation are presented. Numerical simulations of cuspons and periodic cuspons are given to show the correctness of our results.

Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

“…(1.2) by a transformation formula. Secondly, based on the bifurcation method of dynamical systems [8][9][10][11][12][13], the exact traveling wave solutions of Eq. (1.2) are given.…”

Infinitely many solitary waves of an integrable equation with singularity

“…In 2014, Pan et al [16] proposed a completely integrable equation (1.5) which is associated with the KdV equation by the reciprocal transformation. By means of the singular transformation u(x, t) = −1/m(x, t), Eq.…”

“…(1.6) are presented [16]. Little seems to be known about the infinitely many solitary waves and their properties of the completely integrable equations with singularity.…”

“…For this reason, the scope of this paper is to explore the relationship between the existence of the infinitely many solitary waves and the wave speed for Eq. (1.5) by adopting the bifurcation method of dynamical systems [6,7,[16][17][18][19][20][21][22][23]. We list the other approaches for solving the solitary waves for comparison [5,[8][9][10]24,25].…”

Further results on the smooth and nonsmooth solitons of the Novikov equation

“…(1.1). Then, we could apply the bifurcation method of dynamical systems [8,[11][12][13][14][15][16][17][18][19][20][21] further to discuss the dynamical behaviors of the traveling wave solutions and solve the expressions of the smooth and nonsmooth soliton solutions for Eq. (1.1).…”