Often, Vegeta becomes more in touch with his emotions. He starts to care about things other than just fighting, like his family with Bulma. He may show more love and protectiveness towards her and their child.
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In many fanfictions, Vegeta starts as an outsider. He's a Saiyan in a world full of mages. His initial development is about adapting. He has to learn the rules of the Fairy Tail world. For instance, he might not understand the concept of magic guilds at first. As he spends more time there, he becomes more open - minded. He starts to see the value in the friendships and teamwork that are so important in Fairy Tail.
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The relationship could develop in a more emotional way. Maybe Trunks gets injured badly in a fight, and Vegeta realizes how much he cares for his son. He then becomes more protective and affectionate towards Trunks. On the other hand, Trunks could also start to understand his father better, seeing that Vegeta's tough love is actually a way of showing his love and concern. This new understanding could lead to a more harmonious and loving relationship between them.
Vegeta would likely still be his proud and strong - willed self, but with a newfound tenderness towards Bulma. He might be a bit more protective and less quick to anger. Bulma, on the other hand, would be her intelligent and sassy self, always ready to challenge Vegeta in a playful way.
The relationship might develop gradually. Bulma could start by being attracted to Vegeta's power and mystery. As she gets closer to him, she discovers his more complex personality. Vegeta may initially resist her advances but eventually gives in to his own feelings.
In such fanfiction, Vegeta might make Bulma happy by showing his softer side. Maybe he cooks a special meal for her. Vegeta is often portrayed as tough, but when he does something out of his usual character like cooking, it can be a sweet surprise for Bulma.
In the fanfiction, their relationship could be explored by showing how they adapt to each other. Vegeta's feral side might require Bulma to be more patient and understanding. And Bulma's needs might force Vegeta to become more domesticated in some ways. For instance, he might have to learn to control his temper around her family and friends. It could also show how their relationship evolves over time, starting from initial attraction despite their differences to a deep, loving bond that endures through various challenges.