We plot the geometry of several distance-based quantifiers of coherence for Bell-diagonal states. We find that along with both l1 norm and relative entropy of coherence changes continuously from zero to one, their surfaces move from the separable regions to the entangled regions. Based on this fact, it is more illuminating to use an intuitive geometry to explain quantum states with nonzero coherence can be used for entanglement creation, rather than the other way around. We find the necessary and sufficient conditions that quantum discord of Bell-diagonal states equal to its relative entropy of coherence and depict the surfaces of the equality. We give surfaces of relative entropy of coherence for X states. We show the surfaces of dynamics of relative entropy of coherence for Bell-diagonal states under local nondissipative channels and find that all coherence under local nondissipative channels decrease.

We investigate the l1 norm of coherence of quantum states in mutually unbiased bases. We find that the sum of squared l1 norm of coherence of the mixed state single qubit is less than two. We derive the l1 norm of coherence of three classes of X states in nontrivial mutually unbiased bases for 4-dimensional Hilbert space is equal. We proposed "autotensor of mutually unbiased basis(AMUB)" by the tensor of mutually unbiased bases, and depict the level surface of constant the sum of the l1 norm of coherence of Bell-diagonal states in AMUB. We find the l1 norm of coherence of Werner states and isotropic states in AMUB is equal respectively.

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The bounded and smooth solitary wave solutions of 10 nonlinear evolution equations with a positive fractional power term of dependent variable are successfully obtained by homogeneous balance principle and with the aid of sub-ODEs that admits a solution of sech-power or tanh-power type. In the special cases that the fractional power equals to 1 and 2, the solitary wave solutions of more than 10 important model equations arisen from mathematical physics are easily rediscovered.

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