Related papers: Compacton equations and integrability: the Rosenau…

Compactons are compactly supported solitary waves for nondissipative evolution equations with nonlinear dispersion. In applications, these model equations are accompanied by dissipative terms which can be treated as small perturbations. We…

We study the properties of compacton-anticompacton collision processes. We compare and con- trast results for the case of compacton-anticompacton solutions of the K(l, p) Rosenau-Hyman (RH) equation for l = p = 2, with…

Extending a Pade approximant method used for studying compactons in the Rosenau-Hyman (RH) equation, we study the numerical stability of single compactons of the Cooper-Shepard-Sodano (CSS) equation and their pairwise interactions. The CSS…

Canonical coordinates for both the Schroedinger and the nonlinear Schroedinger equations are introduced, making more transparent their Hamiltonian structures. It is shown that the Schroedinger equation, considered as a classical field…

We study a class of generalized fifth order Korteweg-de Vries (KdV) equations which are derivable from a Lagrangian L(p,m,n,l) which has variable powers of the first and second derivatives of the field with powers given by the parameters…

The two matrix spectral problems of Ablowitz-Kaup-Newell-Segur (AKNS) and Kaup-Newell (KN) types associated with so(3,R) are generalized. The corresponding hierarchies of generalized soliton equations are derived by the standard procedure…

In an earlier paper Cooper, Shepard, and Sodano introduced a generalized KdV equation that can exhibit the kinds of compacton solitary waves that were first seen in equations studied by Rosenau and Hyman. This paper considers the…

The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation $\boldsymbol{K} \boldsymbol{M}+\boldsymbol{M} \boldsymbol{K}=\boldsymbol{r}\,…

We present compacton-like solution of the modified KdV equation and compare its properties with those of the compactons and solitons. We further show that, the nonlinear Schr{\"o}dinger equation with a source term and other higher order…

Using a priori estimates we prove that initially nonnegative, smooth and compactly supported solutions of the $K(m,n)$ equations must lose their smoothness in finite time. Formation of a singularity is a prerequisite for the emergence of…

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, \Delta_p u + a(x)u^{p-1} = \lambda f(u,x), (E2) where $\Delta_p$ is the $p-$laplacian, with $1<p<n$. The…

We enlarge the spectral problem of a generalized D-Kaup-Newell (D-KN) spectral problem. Solving the enlarged zero-curvature equations, we produce integrable couplings. A reduction of the spectral matrix leads to a second integrable coupling…

A potential representation for the subset of traveling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves a reduction of a third order partial differential equation to a first order ordinary…

We consider perturbations of the special pole-free joint solution $U(x,t)$ of the Korteweg--de Vries equation $u_t+uu_x+\frac{1}{12}u_{xxx}=0$ and $P_I^2$ equation $u_{xxxx}+10u_x^2+20uu_{xx}+40(u^3-6tu+6x)=0$ under the action of the KdV…

We consider fifth-order nonlinear dispersive $K(m,n,p)$ type equations to study the effect of nonlinear dispersion. Using simple scaling arguments we show, how, instead of the conventional solitary waves like solitons, the interaction of…

Let $(M, g)$ be a closed Riemannian manifold of dimension $n \geq 3$, and let $h \in C^1(M)$ be such that the operator $\Delta_g + h$ is coercive. Fix $x_0 \in M$ and $s \in (0, 2)$. We obtain uniform bounds on the solutions of the critical…

In this paper, we report an interesting integrable equation that has both solitons and kink solutions. The integrable equation we study is $(\frac{-u_{xx}}{u})_{t}=2uu_{x}$, which actually comes from the negative KdV hierarchy and could be…

We present a systematic approach for calculating higher-order derivatives of smooth functions on a uniform grid using Pad\'e approximants. We illustrate our findings by deriving higher-order approximations using traditional second-order…

A weak formulation is devised for the K(m,n) equation which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones…

We consider $u_t=u^{\alpha} u_{xxx}+n(u)u_xu_{xx}+m(u)u_x^3+ r(u)u_{xx} +p(u)u_x^2 + q(u)u_x+s(u)$ with $\alpha=0$ and $\alpha=3$, for those functional forms of $m, n, p, q, r, s$ for which the equation is integrable in the sense of an…