We study the stability of spatially periodic solutions to the Kawahara equation, a fifth order, nonlinear partial differential equation. The equation models the propagation of nonlinear water-waves in the long-wavelength regime, for Weber numbers close to 1/3 where the approximate description through the Korteweg-de Vries (KdV) equation breaks down. Beyond threshold, Weber number larger than 1/...

We consider the generalized integrable fifth order nonlinear Korteweg-de Vries (fKdV) equation. The extended Tanh method has been used rigorously, by computational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact soluti...

A general class of KdV-type wave equations regularized with a convolution-type nonlocality in space is considered. The differs from the nonlinear nonlocal unidirectional previously studied by addition linear convolution term involving third-order derivative. To solve Cauchy problem we propose semi-discrete numerical method based on uniform spatial discretization, that an extension published wor...

We give a unified view of the relation between the SL(2) KdV, the mKdV, and the UrKdV equations through the Fréchet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no nonlocal operators. We extend this method to the SL(3) KdV equation, i.e., Boussinesq(Bsq) equation and obtain the hamiltonian structure of Ur-Bs...

We consider the logarithmic Korteweg–de Vries (log–KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in H(R) with conserved L norm and energy, we construct a weak global solution of the log–KdV equation in a subset of H(R). This construction yields c...

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...

This paper studies the problem of optimal control of the viscous KdV-Burgers’ equation. We develop a technique to utilize the Cole-Hopf transformation to solve an optimal control problem for the viscous KdV-Burgers’ equation. While the viscous KdV-Burgers’ equation is transformed into a simpler linear equation, the performance index is transformed to a complicated rational expression. We show t...