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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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On a system of non-linear differential equations

Consider the following system of coupled differential equations \begin{align} \dot{x}_{1}&= -b_1\sin(x_{1})+c(\sin(x_{2})-\sin(x_{3})) \\ \dot{x}_{2}&= a-2c\sin(x_{2})+b_1\sin(x_{1})-b_4\sin(x_{4}) \\ …
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