We investigate possible linear waves and nonlinear wave interactions in a bounded three-layer fluid system using both analysis and numerical simulations. For sharp interfaces, we obtain analytic solutions for the admissible linear mode-one parent/signature waves that exist in the system. For diffuse interfaces, we compute the overtaking interaction of nonlinear mode-two solitary waves. Mathemat...

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution vorticity. This is a longstanding open problem, even in irrotational case there are only partial results relying on sign conditions or smallness assumptions. As corollary, we obtain relatively complete classification waves: they must be supercritical, symmetric, m...

The Euler-Korteweg model is made of the standard Euler equations for compressible fluids supplemented with the Korteweg tensor, which is intended to take into account capillary effects. For nonmonotone ‘pressure’ laws, the Euler-Korteweg model is known to admit solitary waves, even though their physical significance remains unclear. In fact, several kinds of solitary waves, with various endstat...

A generalized Fisher equation involving a nonlinear term of any order and timedependent coefficients is investigated. We present exact analytical solutions describing periodic and solitary wave solutions using the modified sine-cosine method. A certain class of exact soliton-like solutions has been found by means of the auxiliary equation method. The conditions of existence and uniqueness of th...

Stability and Dynamics of Solitary Waves in Nonlinear Optical Materials

Weak interactions of solitary waves in the generalized nonlinear Schrödinger equations are studied. It is first shown that these interactions exhibit similar fractal dependence on initial conditions for different nonlinearities. Then by using the Karpman-Solov'ev method, a universal system of dynamical equations is derived for the velocities, amplitudes, positions, and phases of interacting sol...

The propagation of temporal solitary waves, formed in the second-order nonlinear medium, is analysed in waveguide directional couplers. The influence of the thirdorder nonlinearity on the switching effect is also presented. Recently, the great effort of nonlinear guided-wave phenomena research is focused on the cascaded second-order process which can produce relatively large intensity-dependent...