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I have a system of two PDEs:

[tex]y_t+(h_0v)_x=0 \quad (1a)[/tex]

[tex]v_t+y_x=0 \quad (1b)[/tex],

where [itex]h_0[/itex] is a constant.

Then I want to show that (1) has traveling wave solutions of the form

[tex]y(x,t)=f(x-ut) \quad (2a)[/tex]

[tex]v(x,t)=g(x-ut) \quad (2b)[/tex],

where [itex]u[/itex] is the propagation velocity.

Differentiating (1a) w.r.t. [itex]x[/itex] and (1b) w.r.t. [itex]t[/itex] I conclude that

[tex]\frac{\partial^2v}{\partial t^2}=h_0\frac{\partial^2v}{\partial x^2}[/tex],

but I am not sure that this is useful. I have tried a linear change of variables in order to arrive at a solution. Introducing the variables [itex]\alpha=ax+bt[/itex] and [itex]\beta=cx+dt[/itex] I arrive at a solution for [itex]v[/itex], namely either [itex]v=C(x+t)[/itex] or [itex]v=C(x-t)[/itex], depending on how the constants [itex]a,b,c,d[/itex] are chosen. Here [itex]C[/itex] is an arbitrary function. Indeed this looks like (2b), but I would like to put forward a more rigorous argument.

I would like some thoughts on this problem, and any hint on how to solve it is welcome. And is it possible to say anything about the relationship between [itex]f[/itex] and [itex]g[/itex] in (2)?

[tex]y_t+(h_0v)_x=0 \quad (1a)[/tex]

[tex]v_t+y_x=0 \quad (1b)[/tex],

where [itex]h_0[/itex] is a constant.

Then I want to show that (1) has traveling wave solutions of the form

[tex]y(x,t)=f(x-ut) \quad (2a)[/tex]

[tex]v(x,t)=g(x-ut) \quad (2b)[/tex],

where [itex]u[/itex] is the propagation velocity.

Differentiating (1a) w.r.t. [itex]x[/itex] and (1b) w.r.t. [itex]t[/itex] I conclude that

[tex]\frac{\partial^2v}{\partial t^2}=h_0\frac{\partial^2v}{\partial x^2}[/tex],

but I am not sure that this is useful. I have tried a linear change of variables in order to arrive at a solution. Introducing the variables [itex]\alpha=ax+bt[/itex] and [itex]\beta=cx+dt[/itex] I arrive at a solution for [itex]v[/itex], namely either [itex]v=C(x+t)[/itex] or [itex]v=C(x-t)[/itex], depending on how the constants [itex]a,b,c,d[/itex] are chosen. Here [itex]C[/itex] is an arbitrary function. Indeed this looks like (2b), but I would like to put forward a more rigorous argument.

I would like some thoughts on this problem, and any hint on how to solve it is welcome. And is it possible to say anything about the relationship between [itex]f[/itex] and [itex]g[/itex] in (2)?

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