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Register a new userTraveling wave solutions for the generalized (2+1)-dimensional Kundu-Mukherjee-Naskar equation
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In this paper, we consider two types of traveling wave systems of the generalized Kundu-Mukherjee-Naskar equation. Firstly, due to the integrity, we obtain their energy functions. Then, the dynamical system method is applied to study bifurcation behaviours of the two types of traveling wave systems to obtain corresponding global phase portraits in different parameter bifurcation sets. According to them, every bounded and unbounded orbits can be identified clearly and investigated carefully which correspond to various traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation exactly. Finally, by integrating along these orbits and calculating some complicated elliptic integral, we obtain all type I and type II traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation without loss.
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We consider the $(1+1)$-dimensional quasilinear wave equation $g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ on $mathbb{R}timesmathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $gin L^{infty}(mathbb{R})$ is even with $g otequiv 0$ and $h(x)=gamma,delta_0(x)$ with $gammain{mathbb{R}}backslash{0}$ and $delta_0$ the delta-distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k=-frac{d^2}{dx^2}-k^2omega^2g$ on $L^2(mathbb{R})$ for all $kin 2mathbb{Z}+1$ together with additional properties of the fundamental set of solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer the problem of finding a suitably defined weak solution to finding a minimizer of a functional on a sequence space. The solutions that we have found are exponentially localized in space. Moreover, we show that they can be well approximated by truncating the Fourier series in time. The guiding examples, where all assumptions are fulfilled, are explicitly given step potentials and periodic step potentials $g$. In these examples we even find infinitely many distinct breathers.
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