Several viruses are known to change their surface antigen types after infecting a host, thereby escaping the immune defence and ensuring persistent infection. In this paper, we theoretically study the pattern of intra-host micro-evolution of pathogen antigen variants under the antigen specific immune response. We assume that the antigen types of the pathogen can be indexed in one-dimensional space, and that a mutation can produce a new antigen variant that is one step distant from the parental type. We also assume that antibodies directed to a specific antigen can also neutralize similar antigen types with a decreased efficiency (cross-reactivity). The model reveals that the pattern of intra-host antigen evolution critically depends on the width of cross-reactivity. If the width of cross-reactivity is narrower than a certain threshold, antigen variants gradually evolve in antigen space as a travelling wave with a constant wave speed, and the total pathogen density approaches a constant. In contrast, if the width of cross-reactivity exceeds the threshold, the travelling wave loses stability and the distribution of antigen variants fluctuates both in time and in genotype space. In the latter case, the expected episodes after infection are a series of intermittent outbreaks of pathogen density, caused by distantly separated antigen types. The implication of the model to intra-host evolution of equine infectious anaemia virus and human immunodeficiency virus is discussed.