Pulse-like spatial patterns described by higher-order model equations
Consider the scalar differential equation \[ \gamma \frac{d^4 u}{dx^4} = \frac{d^2 u}{dx^2} + f(u) \quad\text{for }\gamma > 0, \tag{\(*\)} \] under the assumptions: \((H_1)\) \(f \in C^2 (R,R)\) has at least two zeroes \(s=a\) and \(s=1\) \((a <1)\). \((H_2)\) \(f > 0\) on \((a,1)\), \(f' (a) > 0\), \(f' (1):= - \sigma < 0\). Let \(F(u):= \int^1_u f(s)ds\) and let \(b\) be the infimum of \(s<1\) such that \(F>0\) on \((s,1)\). \((H_3)\) \(-\infty < b < a\), \(f <0\) on \([b,a)\). \((H_4)\) \(f(s) \leq f'(1)(s-1)\) for \(s \in [b,1)\). The authors look for solutions to \((*)\) satisfying \((u,du/dx , d^2 u/dx^2 ,d^3 u/dx^3) \rightarrow (1,0,0,0)\) as \(x\rightarrow \pm \infty\) that is, \(u\) tends to the stable equilibrium point of the system \(du/dt = f(u)\). These homoclinic solutions are called solitons. In particular, the authors look for solitons which are symmetric with respect to \(x=0\). Under the conditions above, the authors prove the existence of even solitons for \(0 < \gamma \leq 1/4\sigma\) by means of a shooting argument and exploiting the property that \((*)\) has a first integral. Moreover, they estimate the amplitude \(u(0)\) and investigate the asymptotic behavior for \(x\rightarrow \pm \infty\). Under additional assumptions, the uniqueness of a symmetric soliton is proven. Furthermore, it is proved that for all \(\gamma > 0\) there is a family of periodic solutions to \((*)\). The obtained results are applied to a generalized nonlinear Schrödinger equation which arises in the theory of pulse propagation through optical fibers.
generalized nonlinear Schrödinger equation
scientific article; zbMATH DE number 1241037