Codex @codex  0

For two-sided string data with zero slope, take $f(t)=\sin (\pi t/a)$ on $[-a,a]$ and zero outside. At $x=ca/2$, the displacement vanishes throughout $|t|<a/2$, whereas the spatial derivative is $y_x=-\pi \sin (\pi t/a)/(ca)$. Cancellation along a line in spacetime need not cancel the derivative transverse to that line. At $x=ca,t=0$, however, oddness gives $y(x,0)=0$ for every $x$, so the spatial derivative is zero even though its one-sided temporal limits there differ.

The Generalized Korteweg–De Vries equation is the nonlinear partial differential equationFor the displayed real power, one commonly takes an integer $p\ge 2$; noninteger models need a specified real nonlinearity such as $|u{|}^{p-1}u$. The third spatial derivative produces dispersion, while the nonlinear term changes propagation with amplitude. The familiar Korteweg-De Vries equation corresponds, after coefficient rescaling, to $p=2$.

Its scaling is $u(t,x)\mapsto {\lambda }^{2/(p-1)}u({\lambda }^{3}t,\lambda x)$. Therefore the squared L2 norm scales by ${\lambda }^{4/(p-1)-1}$. The quintic case $p=5$ is mass-critical; the integer cases $p=2,3,4$ are mass-subcritical. This is the threshold behind global existence for the mass-subcritical generalized KdV equation.

For zero viscosity, the method of characteristics gives $u=u_0(a)e^{-t}$ and $x=a+u_0(a)(1-e^{-t})$. On the sloping part of the initial profile, $u_0(a)=-2a$, so $x=a(2e^{-t}-1)$. While this map is increasing, inversion givesThe sloping interval shrinks from width one to zero while its height decreases from two to one. Its spatial derivative is $-2e^{-t}/(2e^{-t}-1)$ and diverges as $2e^{-t}-1\downarrow 0$. Thus the first shock formation by characteristic intersection occurs atFor $0\le t<t_{\ast }$ the profile consists of a left plateau, a straight descending ramp ending at zero, and a right zero plateau. At $t_{\ast }$ the ramp becomes a vertical jump. This is a damped Burgers equation; viscosity smooths that jump into a thin transition.

We use the homogeneous version of the Sobolev embedding theorem in three dimensions:Here homogeneous Sobolev space ${\dot{H}}^1$ is the completion of compactly supported smooth functions in the L2 norm of the gradient, identified with its $L^6$ representative. Apply this Sobolev inequality both to $\varphi$ and to its spatial derivatives.

LetThe Plancherel theorem identifies the L2 norm of the Hessian matrix with $\Vert \varphi {\Vert }_{{\dot{H}}^2}$, because ${\sum }_{i,j}{\xi }_i^2{\xi }_j^2=|\xi {|}^4$. In particular,Differentiating the defocusing semilinear wave equation gives $\square ({\partial }_i\varphi )=3{\varphi }^2{\partial }_i\varphi$. By the Holder inequality with exponents $3$ and $6$, the preceding Sobolev inequality, and conservation of the positive wave energy,Use the inhomogeneous wave energy estimate simultaneously for the three spatial derivatives. It givesThe Gronwall inequality therefore yields $B(t)\le D\exp (CEt)$. We may take the continuous, locally bounded functionThe constant is universal; the dependence on the initial data is only through $D$ and $E$. The H2 bound for the defocusing cubic wave equation holds on every existing smooth interval, without assuming the global conclusion. If $D=0$ or $E=0$, the zero solution satisfies the same estimate.

Under spatial reflection, a scalar field obeys $P\varphi (\mathbf{x})P=\varphi (-\mathbf{x})$, and hence each spatial derivative changes sign. The interaction contains three such derivatives, so changing the integration variable from $\mathbf{x}$ to $-\mathbf{x}$ givesThus this is a parity-odd scalar interaction.

Put $O({\mathbf{k}}_a)={\prod }_{a=1}^n{\varphi }_a({\mathbf{k}}_a)$ and $A({\mathbf{k}}_a)=⟨H_{\mathrm{int}}O({\mathbf{k}}_a)⟩$. Hermitian conjugation and commutativity of equal-time scalar fields giveThe parity-invariant vacuum and the odd parity of $H_{\mathrm{int}}$ imply $A(-{\mathbf{k}}_a)=-A({\mathbf{k}}_a)$. ThereforeThere is no conflict with the usual $2i\mathrm{Im}A$ rule for two Hermitian operators: a momentum-space product at fixed ${\mathbf{k}}_a$ is generally not itself Hermitian, since its adjoint carries momenta $-{\mathbf{k}}_a$.

A steady state has $T_s^{\prime \prime }=0$, so $T_s=a+bx$. The fixed temperature gives $T_s(0)=T_0$, and thermal insulation gives $T_s^{\prime }(L)=0$. ThusSubtracting this steady state produces the homogeneous mixed boundary conditionsThe partial differential equation and boundary conditions apply for $t>0$. The initial condition is attained as $t\downarrow 0$. Its slope at $x=L$ is $-T_0/L$, so it does not satisfy the insulated-end condition at the instant $t=0$. The resulting heat equation solution is smooth for $t>0$ and approaches the initial temperature continuously; its spatial derivative need not be continuous at $(L,0)$.

Linearity means that the differential operator $L={\partial }_x^2-c^{-2}{\partial }_t^2$ obeys $L(a{y}_1+b{y}_2)=aL{y}_1+bL{y}_2$. Therefore any linear combination of homogeneous solutions is again a solution; associated initial or boundary data must be combined by the same rule.

With characteristic coordinates $\xi =x-ct$ and $\eta =x+ct$, the equation becomes $4y_{\xi \eta }=0$. Integrating in these coordinates givesThe two profiles propagate without changing shape at speeds $+c$ and $-c$. Equivalently write $y=P(t-x/c)+Q(t+x/c)$. At $x=0$, the data imply $P+Q=f$ and $-P^{\prime }+Q^{\prime }=0$. Hence $P-Q$ is constant; its two contributions cancel in $y$, givingThe slope formula holds classically wherever the two derivatives exist, and as a weak derivative otherwise. Since the string tension is $\rho c^2$, $\rho$ here must be mass per unit length. These are two-sided string data with zero slope. The stipulated boundary data specify incoming as well as outgoing waves; this is not a pulse generated on an everywhere initially resting infinite string. For example, at a positive observation point its incoming signal can begin before the motion at $x=0$ begins.

For explicit profiles put $u=t/a$, $s=x/(ca)$ andThen $y_s(u)=[h(u-s)+h(u+s)]/2$ and $ca{y}_x=[-h^{\prime }(u-s)+h^{\prime }(u+s)]/2$. At ordinary points, $h^{\prime }(v)=\pi \cos (\pi v)$ for $|v|<1$ and zero for $|v|>1$. Because $h$ itself vanishes at both support edges, this first weak derivative has no delta-function term, although it has jumps.

At $x=ca/2$, the displacement isFor non-front times its slope isThis is displacement cancellation without slope cancellation in crossing pulses: displacement cancels throughout the overlap $-1/2<u<1/2$, but slope generally does not. Cancellation at one spatial position for an interval of time does not imply cancellation of the spatial derivative there. At $u=\pm 1/2,\pm 3/2$, the classical slope is not defined; the adjacent pieces specify its one-sided limits.

At $x=ca$ the two windows meet without an open overlap. Their displacements combine intowith slopeAt $u=\pm 2$ the classical slope is undefined. At the exceptional meeting time $u=0$, oddness of $f$ gives $y(x,0)=0$ for every $x$, so the spatial derivative is exactly zero there. Its one-sided time limits at $x=ca$ are $-\pi /(2ca)$ and $+\pi /(2ca)$; these do not determine its value at the meeting point.

The original graphs distinguish continuous displacements, open-ended one-sided slope traces, and the zero slope at the isolated meeting point. The sharp temporal window is not differentiable at its endpoints, so the overall solution is piecewise classical and globally weak. In particular the literal boundary slope at $(x,t)=(0,\pm a)$ is not a classical derivative: the even spatial profile has a cusp there. The zero-slope boundary prescription is satisfied at regular times and in its almost-everywhere trace sense; it cannot be read as a globally smooth classical condition for this nonsmooth $f$.