When the char-rnn is trained, the weights of the network is fixed. If I use the same first char, how can I get the different sentence? Such as the two sentences "What is wrong?" and "What can I do for you?" have the same first word "W". Can the char-rnn generate the two different sentences?
Yes, you can get different results from the same state by sampling. Take a look at min-char-rnn by Andrej Karpathy. The sample code is at line 63:
def sample(h, seed_ix, n):
"""
sample a sequence of integers from the model
h is memory state, seed_ix is seed letter for first time step
"""
x = np.zeros((vocab_size, 1))
x[seed_ix] = 1
ixes = []
for t in xrange(n):
h = np.tanh(np.dot(Wxh, x) + np.dot(Whh, h) + bh)
y = np.dot(Why, h) + by
p = np.exp(y) / np.sum(np.exp(y))
ix = np.random.choice(range(vocab_size), p=p.ravel())
x = np.zeros((vocab_size, 1))
x[ix] = 1
ixes.append(ix)
return ixes
Starting from the same hidden vector h and seed char seed_ix, you'll have a deterministic distribution over the next char - p. But the result is random, because the code performs np.random.choice instead of np.argmax. If the distribution is highly peaked at some char, you'll still get the same outcome most of the time, but in most cases several next chars are highly probable and they will be sampled, thus changing the whole generated sequence.
Note that this isn't the only possible sampling procedure: temperature-based sampling is more popular. You can take a look at, for instance, this post for overview.