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In this paper, we are concerned with the existence of solitary waves for a generalized Kawahara equation, which is a model equation describing solitary-wave propagation in media. We obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the generalized Kawahara equation without delay and perturbation by employing the phase space analysis. Furthermore the existence of solitary wave solutions for the equation with two types of special delay convolution kernels is proved by combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality. We also discuss the asymptotic behaviors of traveling wave solutions by means of the asymptotic theory. Finally, some examples are given to illustrate our results.

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Complementary information from multi-modal MRI is widely used in clinical practice for disease diagnosis. Due to scan time limitations, image corruptions, and different acquisition protocols, one or more contrasts may be missing or unusable. Recently developed CNN models for contrast synthesis are unable to capture the intricate dependencies between input contrasts and are not dynamic to the varying number of inputs. This work proposes a novel Multi-contrast and Multi-scale vision Transformer (MMT) that can take any number and combination of input sequences and synthesize the missing contrasts.